Methods and apparatuses for estimating transient slip

ABSTRACT

A method of determining a slip estimate associated with an induction motor through analysis of voltage and current signals. A fundamental frequency is calculated from a representation (e.g., complex representation) of the voltage signal, and a saliency frequency is calculated from a representation of the current signal. An estimation of slip quantity is calculated according to a slip estimation function that includes the saliency frequency, a saliency order, the fundamental frequency, a quantity of rotor slots, and a quantity of poles of the motor.

CROSS-REFERENCE TO RELATED APPLICATION

This application claims the benefit of U.S. Provisional Application Ser.No. 61/053,941, filed May 16, 2008, entitled “Methods and Apparatusesfor Estimating Transient Slip, Rotor Slots, Motor Parameters, andInduction Motor Temperature.”

FIELD OF THE INVENTION

The present invention relates generally to the estimation of transientslip of a motor and, in particular, to methods for passive temporalestimation of the transient slip of a motor.

BACKGROUND

Robust accurate transient slip estimation is fundamental to achieving asignificant measure of success in several types of motor analysis.Established techniques often explicitly or implicitly assume stationaryor quasi-stationary motor operation. Practically, motor and loadenvironments are dynamic, and dependent on myriad variables includinginternal and external temperature, and varying load dynamics,fundamental frequency and voltage. The cumulative effects of suchvariations can have significant detrimental effect on many types ofmotor analysis.

Model Referencing Adaptive System (MRAS) is a method of iterativelyadapting an electrical model of a three-phase induction motor withsignificant performance advantages over competitive approaches assumingquasi-stationary motor operation, as the assumption of stationaryoperation is often violated, with detrimental effect on model accuracy.MRAS is highly dependent upon availability of robust transient slipestimation. Applications benefiting from accurate transient slipestimation include, but are not limited to, synthesis of high qualityelectrical and thermal motor models, precision electrical speedestimation, dynamic efficiency and output power estimation, andinverter-fed induction machines employing vector control.

Slip estimation can be performed by passive analysis of the voltage andcurrent signals of a three-phase induction motor, and commonly availablemotor-specific parameters. In a stationary environment, slip estimationaccuracy is optimized by increasing the resolution of the frequency, ordegree of certainty of the frequency estimate. As motor operation istypically not stationary, robust slip estimation also depends uponretention of sufficient temporal resolution, such that the unbiasedtransient nature of the signal is observed.

Frequency estimation is commonly performed by Fourier analysis. Fourieranalysis is frame-based, operating on a contiguous temporal signalsequence defined over a fixed period of observation. Frequencyresolution, defined as the inverse of the period of observation, can beimproved by extending the period of observation, and moderateimprovement in effective frequency resolution can be realized throughlocal frequency interpolation and other techniques that are generallydependent on a priori knowledge of the nature of the observed signal.

In Fourier analysis, it is implicit that source signals are stationaryover the period of observation, or practically, that they are stationaryover contiguous temporal sequences that commonly exceed the period ofobservation in some statistical sense. Fourier analysis can beinappropriate for application in environments where a signal of interestviolates the stationary condition implicit in the definition of aspecific period of observation, resulting in an aggregate frequencyresponse and loss of transient information defined to be important toanalysis.

Fourier analysis presents a temporal range versus frequency resolutiondilemma. At periods of observation to resolve a saliency harmonicfrequency with sufficient accuracy, the stationary condition is violatedand important transient information is not observable.

An effective saliency harmonic frequency resolution of less than 0.1 Hzcan be defined to be minimally sufficient, resulting in a requisiteperiod of observation of 10 seconds, which can be reduced to no lessthan 1.0 second through application of local frequency interpolation.However, saliency harmonic frequency variations exceeding 10.0 Hz arenot uncommon. The relative transient nature of such signals violates thestationary condition requirement implicit in the definition of theminimum period of observation. Fourier analysis is not suitable foremployment in robust accurate slip estimation, in applications wherepreservation of significant transient temporal response is important.

Thus, there is a need for a method of transient slip estimation based onpassive temporal analysis, which results in robust accurate slipestimation with superior temporal resolution relative to competitivemethods.

BRIEF SUMMARY

A method for estimating the transient slip of a motor based on passivetemporal analysis is described. The method results in a robust andaccurate slip estimation with superior transient response relative tocompetitive methods, as important information defining the dynamicnature of the motor system is preserved, while retaining the accuracy tosupport advanced motor modeling and analysis applications.

Motor voltage and current signals are acquired in the form of sampledthree-phase sequences, with one phase or dimension per source line. Thesampled sequences are converted to complex representations throughapplication of simple linear transformations. The complex conversion canbe equivalent, reversible, lossless, mathematically convenient, and/orpractically advantageous, relative to three-phase representations, interms of subsequent definitions.

A single phase current signal has conventionally served as the basis formotor analysis, including slip estimation. Each phase in a motor voltageor current signal is unique and contains information not necessarilyobservable in analysis of any other specific single phase. Analysis ofthree-phase signals, or equivalent complex representations, isadvantageous in this context. Complex conversion of a single phasesignal is possible, though complex conjugate symmetry is implicit.Complex conjugate symmetry is not observable in an analysis of thecomplex motor voltage and current signals. Single phase analysisrepresents a loss of information due to the assumption of conjugatesymmetry. A loss of such information is unrecoverable and can besignificant.

According to some aspects, the transient slip estimation method performsall processing on complex signals extracted from acquired three-phasevoltage and current signals. The complex analysis of the describedmethod allows dominant saliency (speed-related) harmonic identificationto consider positive and negative sequence saliency harmonicsindependently, resulting in the identification of the highest qualityobservable saliency harmonic, improving subsequent saliency frequencyestimation accuracy.

The complex voltage and current signals can be decomposed intofundamental and residual components in a process of signal extraction.Complex adaptive filters such as, for example, Complex Single Frequency(CSF) filters, can be applied to extract complex exponential signalcomponents proximate to the rated fundamental frequency, accuratelymatching magnitude and phase. The complex analysis is preferablyutilized in static and adaptive filters employed in transient slipestimation algorithms because performance is degraded when processingcomplex signals in real systems due to loss of information resultingfrom projection onto a real axis, effectively reducing the dimension ofthe signals.

The complex fundamental voltage and current signals are combined tosynthesize an instantaneous input power estimate, which enables robustsaliency frequency and approximate slip estimation. The approximate slipis derived from a rated slip and a normalized input power, which issignificantly more linear and accurate than normalized current-basedmethods.

The fundamental frequency is dynamic, due to variations in the supplyvoltage and load. Though a nominal rated fundamental frequency isavailable, an accurate instantaneous fundamental frequency estimate canbe extracted, and can significantly improve the accuracy of transientslip estimation. The complex voltage is selected as a preferred sourcefor analysis, as it has a relatively high signal to noise ratio, interms of the complex fundamental and residual voltage components. Anysuitable means for estimating the fundamental frequency can be employed.According to some aspects, the fundamental frequency is extractedthrough application of a Phase Locked Loop (PLL). Although thefundamental frequency is a transient signal, it generally varies slowlyrelative to the PLL bandwidth.

Thermal modeling can be employed to estimate the rotor temperature as afunction of the rotor resistance, which is highly dependent upon theslip estimation. The rotor temperature estimates based on slip estimatesextracted with the assumption of static fundamental frequency were foundto have several degrees of error, which was directly and entirelycorrelated with the failure to account for dynamic fundamentalfrequency.

Saliency harmonics present in the complex residual current can beidentified and evaluated to define the highest quality observable, ordominant, saliency harmonic. The saliency harmonic magnitude andrelative proximate noise levels vary with motor geometry and loadconditions. The bandwidth constraints imposed by limiting the samplingfrequency to a minimally sufficient practical rate reduce the range ofobservable saliency harmonics. Of the set of observable saliencyharmonics, the frequency and range corresponding to a dominant saliencyharmonic is desirably estimated.

The dominant saliency harmonic is identified through application of atemporal analysis method, iterating over a limited subset of frequencybands of interest. The process consists of demodulating each potentialsaliency harmonic to extract a saliency frequency. The magnitude of eachremaining identified saliency harmonic is compared to select thedominant saliency harmonic. Identification of the dominant saliencyharmonic results in retention of a dominant saliency frequency and adominant saliency order, and design and retention of saliency filtercoefficients.

The saliency frequency is defined as the instantaneous frequencyestimate of the highest quality observable saliency harmonic duringmotor operation in dynamic conditions. The saliency frequency isestimated from the saliency harmonic, and dependent upon thecorresponding saliency order. Demodulation applies a Voltage ControlledOscillator (VCO) to mix the dominant saliency frequency in the complexresidual current to complex baseband, or zero frequency. A FiniteImpulse Response (FIR) or an Infinite Impulse Response (IIR) filter isapplied to band limit the mixed current, producing a complex basebandcurrent. To complete the demodulation process, a residual frequencycontained in the complex baseband current is extracted, resulting in anaccurate saliency frequency estimate. Alternative methods of iterativefrequency estimation described include Direct, Phase Discriminator (PD),and PLL analysis.

Demodulation initially removes the dominant saliency frequency, which isdefined as the saliency frequency expected during motor operation inrated conditions, from the complex residual current, resulting in acomplex baseband current. Demodulation is completed by estimating theresidual frequency in the complex baseband current, resulting in anaccurate estimate of the saliency frequency. The saliency frequency isdefined as the sum of the dominant saliency frequency and the residualsaliency frequency. Dynamic fundamental frequency and load conditionsare reconciled entirely in the process of iterative frequencyestimation.

The transient frequency estimation of a complex baseband current signalis analogous to FM demodulation. Definition of minimum frequencytracking rate and bandwidth, available resources including computationalcomplexity and bandwidth, and precision called for by the application,can be employed to select an appropriate method of frequency estimationamong myriad possibilities.

The Direct, PD, and PLL methods of estimating the saliency harmonicfrequency are described. These diverse methods provide the flexibilityto generally increase precision at the expense of computationalcomplexity and frequency tracking rate, increasing the practicalapplicability of the transient slip estimation method.

Saliency harmonic and fundamental frequency estimation paths should beexamined to reconcile differences in latency resulting from asymmetricprocessing paths. Latencies are principally contributed by filteroperations, and can readily be estimated, though delays associated withadaptive elements are dependent upon adaptive parameters and call foradditional analysis or experimental quantification.

The complex baseband processing is advantageous in several contexts,including simplicity of design, computational complexity, componentreuse, and performance. The selection of the dominant saliency frequencyto represent the center frequency in an isolated complex basebandcurrent signal calibrates frequency estimation to the nominal ratedconditions. Specific design of the saliency low-pass filter to supportthe bandwidth of the saliency harmonic in operating conditionscorresponding to a specific range of interest, or expected use, resultsin optimal interference rejection from proximate signals which wouldotherwise introduce aggregate estimation error.

Slip estimation can be directly expressed though a reorganization of thesaliency frequency equation, based on availability of accurate saliencyfrequency and fundamental frequency estimation. Thus, the slip isspecified as a function of saliency frequency, fundamental frequency,and motor geometry.

Rotor slots are a static measure of motor geometry which is generallynot known, but can be defined by manufacturer data, direct examination,or analysis of electrical signals and motor parameters. Electricalanalysis provides the most practical solution for rotor slotsestimation, as manufacturer data is not readily available for allmotors, and direct observation is intrusive and time-consuming.

Methods are described for automatically estimating the rotor slots viaelectrical analysis. A practical range of rotor slots can be defined andslip can be iteratively estimated over this range.

Relative to a slip estimation based on the saliency harmonic analysispreviously described, the approximate slip is less precise, though it issufficiently accurate to provide a reference, independent of motorgeometry. Approximate slip can be defined as a function of normalizedpower, and temperature compensated to improve accuracy.

A rotor slots performance surface defines the slip estimation errorbetween approximate slip, and each instance of slip estimation, as afunction of rotor slots. The rotor slots performance surface defines therelative proximity of accurate slip estimation, based on the assumptionof a specific rotor slots solution, and independent approximate slip.The rotor slots performance surface is nonlinear, and a concisedifferentiable representation is unavailable, so the surface is revealedthrough an iterative process of assuming a specific rotor slotssolution, extracting the dominant saliency harmonic and slip estimationbased on this assumption, and quantifying the resulting slip estimationerror.

The rotor slots performance surface has an abscissa of a contiguoussequence of integer rotor slots indices, and an ordinate specified bythe L1 slip estimation error, or mean absolute difference betweenapproximate slip and slip estimates. Rotor slots are estimated bydetermining the global minimum of the rotor slots performance surface,evaluated over a practical rotor slots index range. It is possible toreduce the range of rotor slots evaluated, if the identification of alocal minimum is persistent over a sufficient rotor slots range to beconsidered a probable global minimum.

The rotor slots performance surfaces can vary based on the load andthermal conditions associated with the complex residual current, thefundamental frequency and the approximate slip sequences used to producethem. To ensure robust rotor slots estimation, several rotor slotsestimates are produced from independent rotor slots performance surfacesformed under diverse load and thermal conditions.

A consensus rotor slots estimate can be extracted when the rotor slotsset is sufficiently populated. If a consensus rotor slots estimate isnot available, additional independent rotor slots estimates can be addedto the set until a consensus is available.

The rotor slots are independently estimated from the complex residualcurrent, the fundamental frequency and the approximate slip sequencescorresponding to some diversity of load or thermal conditions beforeidentifying rotor slots with sufficient confidence to decline furtheranalysis. In the event that a consensus rotor slot estimate is notidentified, an optional probabilistic method is described to selectrotor slots from a conflicting rotor slots set, based upon relativeconditional probability.

The probabilistic method extracts a specific conditional ProbabilityDensity Function (PDF) from a three-dimensional matrix of stored PDFsindexed by poles and normalized rated input power. The PDF, a discretefunction of probability as a function of rotor slots index, is queriedfor members of the rotor slots set. The rotor slots estimate is equal tothe rotor slots set member with the highest probability. The PDF matrixis synthesized offline from a motor database.

A method of rotor slots estimation based on frequency estimation of theeccentricity harmonic, proximate to the fundamental frequency, wasproposed by Hurst, and others. The eccentricity method has severaldisadvantages, relative to saliency harmonic analysis, and the temporalmethods and architectures proposed.

Eccentricity harmonics are observable in motor current signals near thefundamental frequency, at some frequency offset related to polequantity. Line-related harmonics and load-related harmonics dominate thelower frequency range, resulting in significant sources of interference.Saliency harmonics are observable at higher frequencies, with lessinterference and generally higher signal quality. Saliency frequencyestimation is enhanced by dynamic selection of the dominant, or highestquality, saliency harmonic observable, and selection of optimalband-limiting filters based on motor geometry, and temporal analysismethod and architectures.

Practical motor environments commonly demonstrate quasi-stationaryoperation on the order of 20 milliseconds. Fourier methods of frequencyestimation call for relatively long periods of observation to providethe frequency resolution to support rotor slots estimation, typically 30to 100 seconds. The implicit assumption that motor operation isstationary over the period of observation is often violated, resultingin inaccurate aggregate frequency estimation. Temporal analysis methodsextract accurate frequency estimation with superior transient response,and are well-suited for application in motor harmonic analysis.

The temporal saliency frequency estimation methods and architecturesdescribed are capable of having sufficient bandwidth to operatecontinuously in diverse practical motor environments, and providingaccurate iterative instantaneous frequency estimates to support robustrotor slots and slip estimation.

The concept of specifying a rotor slots performance surface, derivedfrom independent slip estimates, as a function of rotor slots, anddefining the rotor slots solution at the global minimum of the surfaceis useful.

The means of providing both the approximate slip and transient slipestimates, and the accuracy of these estimates relative to competitivetechnologies improves the accuracy and utility of the rotor slotsperformance surface, and rotor slots estimation.

The method of defining a consensus rotor slots solution from a set ofindependent estimates, extracted by passive observation of diverse loadand thermal conditions, supports robust rotor slots estimation.

The probabilistic method defines a rotor slots solution in the eventthat a consensus estimate is unavailable ensures increases utility andfacilitates broader application of the rotor slots estimation algorithm.

According to alternative aspects, the slip estimation can be estimatedby alternative means including, for example, Fourier analysis orapplication of a multi-rate method such as the zoom FFT. Thesealternative methods are inferior to the transient slip estimation methodpreviously described, as they lack the temporal resolution to supportdependent applications including Model Referencing Adaptive System(MRAS), or transient power factor estimation, they could be used tosynthesize a rotor slots performance surface. The resulting surfacewould have less accuracy, due to the impractical dependence onstationary condition implicit in Fourier analysis, though in manyspecific motor environments, application of alternative slip estimationtechniques can be sufficient to extract a rotor slots estimate.

The definition of alternative slip estimations for use in rotor slotsperformance surface synthesis should be carefully considered, as shouldthe application of temporal methods and architectures to produce thehigh quality slip estimates similar to those described herein.

The described methods of transient slip and rotor slot estimation caninvolve an application of instantaneous frequency estimation based onnovel Phase Discriminator (PD) and Phase Locked Loop (PLL) architecturesand means of adaptation. For example, the transient slip methods involveestimations of instantaneous fundamental frequency and saliency harmonicfrequencies. Additional applications benefiting from the employment ofthe complex PLL or PD are extensive and diverse, including control andcommunications topics such as motion control and FM demodulation.

The Phase Discriminator (PD) is an adaptive filter which estimates theinstantaneous frequency of a primary signal through a process of sourcenormalization to unity magnitude, evaluation of the difference betweenpresent and unity delayed and scaled normalized samples, and adaptationof the complex coefficient which forms the basis of the scale factor tominimize the difference, or error.

Normalization preserves phase, while diminishes the effect of magnitudevariation in the primary signal. An optimum complex coefficient valuecan be found to minimize the resulting error signal, in a least squaressense, by encoding the phase difference between sequential normalizedsamples in the complex coefficient phase. Normalized magnitudes areunity, and the complex coefficient simply defines the phase shift toreconcile change in the sequence. The instantaneous frequency of thesource can be directly estimated by examining the phase of the complexcoefficient.

The error is minimized when the phase of the complex coefficientapproximates the change in phase between adjacent samples in the primarysignal. This change in phase is, by definition, equal to instantaneousfrequency. As the primary signal frequency changes, the phase of thecomplex coefficient adapts to minimize estimation error. As thefrequency generally changes slowly with respect to the bandwidth of thePD, estimation error can be minimized.

The Phase Locked Loop (PLL) is a closed loop adaptive filter optimallysuited to accurately estimate frequency in a dynamic environment withsignificant in-band interference. The complex PLL supports the dynamicidentification and instantaneous frequency estimation of a complexexponential component of a complex signal, in a flexible andcomputationally efficient form. The complex PLL architecture and meansof adaptation have broad applicability in many application domains.

PLL architecture is defined to consist of a VCO, a mixer, an optionalIIR filter, a PD, and a means of frequency adaptation. The PLL elementsare combined to synthesize a phase-contiguous complex exponential at anadaptive instantaneous frequency, such that the mixed signal product ofthe synthesized signal and complex residual current is at complexbaseband, an IIR filter band limits the resulting complex basebandsignal, and a PD estimates the residual frequency, or frequency of thecomplex baseband signal. Complex baseband is a convenient representationof a complex signal whose carrier frequency, or significant complexexponential component of interest, is mixed to nominal zero frequency,as a matter of convenience.

The VCO frequency is iteratively adapted to improve on the estimate ofthe instantaneous frequency to force the complex exponential componentof interest in the complex baseband signal to move to and remain atcomplex baseband, an operation analogous to demodulation of an FMsignal. PD estimates the residual frequency, due to frequency estimationerror or frequency drift, of the signal at complex baseband. PD residualfrequency is employed as an error metric, to modify the VCO frequency.

The convergence of a Least Mean Squared (LMS) means of adaptation isproportional to the rate of adaptation, and inversely proportional tomisadjustment, or noise introduced by the adaptive process. Throughjudicious selection of adaptive parameters, convergence time can bereduced and bandwidth increased, at the expense of increased estimationnoise. To ensure stability and minimize misadjustment, instantaneousfrequency estimation should change slowly, relative to magnitude andphase adaptation in the PD.

PLL filter bandwidth can be defined according to the nature of thecomplex baseband current environment. Bandwidth can be increased, inreturn for significant reduction in latency and improved frequencytracking rate, at the cost of increased aggregate frequency estimationerror. Unity bandwidth selection, which can be appropriate inenvironments with limited in-band interference, effectively excises theIIR filter, eliminating latency contributions by the filter andmaximizing frequency tracking rate

Model Referencing Adaptive System (MRAS) is a method of iterativelyadapting an electrical model of a three-phase induction motor withsignificant performance advantages over competitive approaches assumingquasi-stationary motor operation, as the assumption of stationaryoperation is often violated, with detrimental effect on model accuracy.MRAS is highly dependent upon availability of robust transient slipestimation, which is most effective when employing PD and PLL frequencyestimation methods.

Applications benefiting from accurate transient slip estimation include,but are not limited to, synthesis of high quality electrical and thermalmotor models, precision electrical speed estimation, dynamic efficiencyand output power estimation, and inverter-fed induction machinesemploying vector control.

Applications benefiting from employment of the complex PLL or PD areextensive and diverse, including accurate transient frequencyestimation, and control and communications topics, including motioncontrol and FM demodulation.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing and other advantages of the invention will become apparentupon reading the following detailed description and upon reference tothe drawings.

FIG. 1 is a data flow diagram for estimating the transient slip of amotor.

FIG. 2 is a data flow diagram of a Data Acquisition Process of FIG. 1.

FIG. 3 is a data flow diagram of a Signal Extraction Process of FIG. 1.

FIG. 4 is a diagram of a Latency Compensation Process of FIG. 3.

FIG. 5 is a chart depicting the complex input power estimation for a 15HP, 6 pole motor operating in rated voltage of 460 V and dynamic currentconditions of 10.0-18.5 A, for a period of 60.0 seconds, at a samplingfrequency of 10 kHz.

FIG. 6 is a flow chart diagram of an architecture for estimating afundamental frequency according to some aspects of the implementations.

FIG. 7 is a chart depicting the normalized fundamental frequencyestimation for a 15 HP, 6 pole motor operating in near rated conditions,for a period of 60.0 seconds, at a sampling frequency of 10 kHz.

FIG. 8 is a data flow diagram of a Mechanical Analysis of FIG. 1.

FIG. 9 is a control flow diagram of a Mechanical Analysis of FIG. 8.

FIGS. 10 a and 10 b collectively are a flow chart diagram of anarchitecture for identifying a dominant saliency harmonic.

FIG. 11 is a chart depicting saliency harmonic frequency responsescorresponding to a plurality of complex baseband currents of a motor.

FIG. 12 is a chart depicting a transient residual saliency frequencyestimation for a synthetic complex baseband current signal, for a periodof 10.0 seconds, at a sampling frequency of 10 kHz.

FIG. 13 is a flow chart diagram of an architecture for estimating asaliency frequency according to some aspects.

FIG. 14 is a chart depicting saliency frequency estimations, accordingto various aspects, for a motor operating in discontinuous loadconditions.

FIG. 15 is a chart depicting slip estimations, estimated according tovarious aspects, for a motor operating in discontinuous load conditions.

FIG. 16 is a control flow diagram of a Rotor Slots Process of FIG. 8 andFIG. 9.

FIG. 17 is a chart depicting a slip error, estimated for a motoroperating in continuously increasing, near rated, load conditions.

FIG. 18 is a chart depicting a rotor slots performance surface for amotor operating in continuously increasing, near rated, load conditions.

FIG. 19 is a chart of a Probability Density Function depicting theconditional probability that a motor has a particular rotor slotsquantity.

FIG. 20 is diagram of a Rotor Slots Probability Density Functionarchitecture according to some aspects.

FIG. 21 is a flow chart diagram of an architecture for an infiniteimpulse response (IIR) filter according to some aspects.

FIG. 22 is a flow chart diagram of an architecture for a VoltageControlled Oscillator (VCO) according to some aspects.

FIG. 23 is a flow chart diagram of an architecture for a Complex SingleFrequency (CSF) filter according to some aspects.

FIG. 24 is a flow chart diagram of an architecture for a PhaseDiscriminator (PD) according to some aspects.

FIG. 25 is a flow chart diagram of an architecture for a Phase LockedLoop (PLL) according to some aspects.

While the invention is susceptible to various modifications andalternative forms, specific embodiments have been shown by way ofexample in the drawings and will be described in detail herein. Itshould be understood, however, that the invention is not intended to belimited to the particular forms disclosed. Rather, the invention is tocover all modifications, equivalents, and alternatives falling withinthe spirit and scope of the invention as defined by the appended claims.

DETAILED DESCRIPTION

A method of estimating the transient slip of a motor based onapplication of communication and adaptive systems theory and novelarchitectures to identify and iteratively estimate saliency harmonicfrequency in dynamic motor environments is described. The performance ofthe transient slip estimation methods disclosed are superior to that ofcompetitive slip estimation methods, as important information definingthe dynamic nature of a motor system is preserved, while retaining theaccuracy desired to support advanced motor modeling and analysisapplications.

The fundamental design problem overcome was to identify and accurately,and iteratively, estimate the instantaneous frequency of a high qualitysaliency harmonic, with sufficient bandwidth to preserve the importanttemporal response of the motor system. A saliency harmonic is defined asa speed-related harmonic signal as opposed to line-related orload-related harmonic signal. Similarly, a saliency frequency is definedas a speed-related frequency.

Saliency harmonics are not stationary, though at a single observableinstance multiple saliency harmonics can be found through analysis ofstator current at a common offset, or modulation, relative to proximateodd fundamental frequency harmonics. This common instantaneous saliencyharmonic modulation can be exploited to discriminate and identifysaliency harmonics with respect to line and load-related harmonics.Dynamically, saliency harmonic modulation increases approximatelylinearly with input power, and slip. Secondary effects on saliencymodulation include temperature, as in the absence of thermodynamicequilibrium, slip increases with temperature.

The saliency harmonic equation relates various motor characteristics orquantities such as slip, fundamental frequency, saliency harmonicfrequency, odd-phase harmonic order, rotor slot quantity and polequantity, in a linear relationship. The slip can be approximated byapplying knowledge of the normalized power, the rated power factor andthe rated speed, and additional motor parameters directly extracted fromthe motor name plate, or through analysis derived from commonlyavailable information, supporting prediction of a set of possiblesaliency harmonic frequencies. Motor geometry and design considerationsgenerally conspire to define one saliency harmonic with higher quality,or signal to noise ratio, and to render other potential saliencyharmonics to be either lower quality or unobservable. A priori knowledgemay not be exploited to predefine the most useful saliency harmonicfrequency range for a specific motor. It is determined experimentally.

Saliency harmonics observable in complex residual stator current can beviewed as a set of Frequency Modulated (FM) signals about carrierfrequencies which can be nominally defined as a function of motorgeometry and rated operation conditions. Carrier drift due to dynamicfundamental frequency and temperature, significant in-band interferencedue to line and load-related harmonics, motor-specific dependence of themodulation index, or rate of frequency shift as a function of load, andselection of the optimum carrier frequency are potential problemsspecific to motor analysis which are resolved by the transient slipestimation methods described below.

A useful representation of the interactions in a complex system involvesdecomposing the system into collections defined by logical partitions,and describing the data elements produced and consumed by variousentities across partition boundaries.

Data flow diagrams share a common graphical vocabulary. Collections aredelimited by encapsulating rectangles, containing actors consisting ofexternal observable entities, processes or internal observable entitiesdefined by rounded rectangles, or dynamic or constant data elements. Allcomponents are labeled, and data elements are enumerated with uniquesymbols used consistently in subsequent descriptions. Data flow isexplicitly defined by arrows, indicating the production source, actors,processes, or collections thereof, and the data element or collectionavailable for subsequent consumption.

Control flow specification is desired to define conditional orstate-dependent process sequences, if decision-based definitions arerequired. Absent explicit control flow specification, it is implicitthat control flow is based on data dependencies. Valid implementationscan represent myriad disparate architectures, which are functionallyequivalent.

1.0 Overview of the Transient Slip Estimation Data Flow

Referring to FIG. 1, a transient slip estimation data flow 100,according to some aspects of the implementations, consists of anEnvironment collection 110, an Analog collection 114 and a Digitalcollection 102.

The Environment collection 110 consists of a Supply component 148, aMotor component 150 and a Load component 152. These components areexternal actors, in the context of transient slip estimation, and areaccessible strictly in the sense of passive observation.

The Analog collection 114 consists of a Filter component 156, an Analogto Digital Conversion (ADC) component 154, and a Sensor component 158.These components are precisely internal, though the requirement tocontrol the Analog components is limited, based on information that isgenerally statically defined, and independent of the Environmentcollection 110. It is convenient to consider the components of theAnalog collection to consist of external actors.

The Digital collection 102 consists of a set of internal componentsdefined at the highest level of abstraction, with functionaldecomposition, as a Data Acquisition process 118, a Signal Extractionprocess 120, and a Mechanical Analysis process 122. The term process isnot strictly intended to define a reference design, but rather todescribe a convenient partition of the dynamic consumption andproduction of related data elements.

The Data Acquisition process 118 converts a three-phase voltage signal144 and a three-phase current signal 146 produced by the Analogcollection 114 into a complex voltage representation 124 and a complexcurrent representation 126. The Data Acquisition process 118 will bedescribed in further detail with respect to FIG. 2.

The Signal Extraction process 120 filters the complex voltage 124 andthe complex current 126 signals and extracts a data collection includingfrequency-dependent components such as, a complex input power 134, afundamental frequency 136, and an approximate slip 138. The SignalExtraction process 120 will be described in further detail with respectto FIGS. 3-7.

The Mechanical Analysis process 122 defines a rotor slots quantity 142of the motor 150, and extracts robust, accurate, transient estimates ofslip 140. The Mechanical Analysis will be described in further detailwith respect to FIGS. 8-20.

The transient slip estimation data flow 100 is a function of thethree-phase voltage 144 and the three-phase current 146, and a set ofsystem scalar data 112, generally consisting of motor-specificparameters available from the motor name plate, with the exceptions of aneutral reference 170, an initial rotor temperature 174, and a samplingfrequency 176.

2.0 The Environment Collection

The Environment collection 110 consists of the Supply 148, the Motor 150and the Load 152 components. Direct interaction with the Analogcollection 114 is represented as an unspecified data flow, consisting ofthe raw three-phase voltage 144 and three-phase current 146 signals,prior to quantization. The Environment is concisely described through arelevant collection of static constant data elements 112 with externalvisibility.

The Supply 148 produces the voltage source consumed by the Motor 150.The voltage source includes AC three-phase mains sources rated at, forexample, 208-230, 380, or 460 Volts. The rated fundamental frequency 166of the Motor 150 is, for example, 60 or 50 Hz. The actual fundamentalfrequency 136 is not strictly stationary, and normal fluctuation due tochanges in the Supply 148 or the Load 152 is acceptable.

The Motor 150 represents a three-phase induction motor compatible withdefined Supply 148 requirements, with a rated output power in the rangeof up to 500 HP, or approximately 375 kW.

The Load 152 represents the apparatus driven by the Motor 150.Non-limiting examples of diverse Load classes include conveyors,crushers, cutters, compressors, desiccants, generators, pumps,rotational drives, and the like.

The Supply 148, the Motor 150 and the Load 152 determine the transientand harmonic characteristics of the motor current and power.

The Constants collection 112 includes the information for successfuloperation of the transient slip estimation system 100. The Constant data112 is available at the epoch of system operation, and is extracteddirectly or indirectly from information provided by the motormanufacturer on the name plate of a specific motor 150, or otherwisereadily obtained.

As used herein, rated operation refers to a state of motor operationexplicitly defined by a specified rated voltage 160, rated current 162,and rated fundamental frequency 166. The stator current, the rotorspeed, and the motor power factor are implicitly affected by ambientrotor temperature, and the rated values for these parameters assumestationary operation at a temperature which may not be explicitlyspecified by the motor manufacturer. Analysis of experimental data,motivated by the need to define temperature compensation to calibratecertain motor parameters, supports the assertion that rated temperatureoperation is approximately 55° C. above the ambient temperature.

The Rated Voltage 160, v₀, is the root mean square (“RMS”) Supplyvoltage necessary for motor operation in rated conditions, in units ofVolts.

The Rated Current 162, i₀, is the RMS stator current resulting frommotor operation in rated conditions, in units of Amps.

The Rated Speed 164, r₀, is the rotor speed resulting from motoroperation in rated conditions, in units of revolutions per minute(“RPM”).

The Rated Fundamental Frequency 166, f₀, is the Supply frequencynecessary for motor operation in rated conditions, in units of Hz.

The Rated Power Factor 168, P_(F), is the power factor for motoroperation in rated conditions, in (Equation 1). Some motor name platedefinitions may not explicitly define the rated power factor 168, thoughan estimate can be extracted from available information, includingefficiency, P_(E), rated power, P_(H), in units of horsepower, the ratedvoltage 160 and the rated current 162.

$\begin{matrix}{P_{F} = {P_{E}^{- 1} \cdot \left( \frac{P_{H} \cdot 746}{v_{0} \cdot i_{0}} \right)}} & \left( {{Equation}\mspace{14mu} 1} \right)\end{matrix}$

The Neutral Reference 170, N_(R), describes a configuration of voltagesensors used in the application, in the form of a Boolean variable. Thevoltage sensor topologies can be either line-neutral or line-linereference, resulting in a neutral reference state of true or false,respectively.

The Poles Quantity (Poles) 172, P, for a specific motor 150 aregenerally not specified on the motor name plate, but can be triviallyinferred as a function of the rated speed 164 and the rated fundamentalfrequency 166, in (Equation 2).

$\begin{matrix}{P =_{FLOOR}\left( \frac{f_{0} \cdot 120}{r_{0}} \right)} & \left( {{Equation}\mspace{14mu} 2} \right)\end{matrix}$

The Initial Rotor Temperature 174, Θ_(TR,0), is the ambient temperatureof a motor 150 that is in thermal equilibrium with its surroundings,prior to operation. The initial rotor temperature 174 is used tocalibrate the offset of rotor and stator temperature estimation, asestimated changes in rotor resistance correspond to relative, notabsolute, changes in temperature. To determine absolute rotortemperature, the epoch of operation, or first start, should occur underknown ambient thermal conditions.

A Sampling Frequency 176, f_(S), is the rate at which the analog voltagesignal and the analog current signal are converted to discrete sampledrepresentations 144 and 146, in units of Hz.

The rated fundamental frequency 166 and the sampling frequency 176defined in the Environment collection 110 are specified in units of Hz,a measure of absolute frequency. Unless otherwise explicitly stated, allother frequency definitions are specified in terms of normalizedfrequency. The normalized frequency, f, is a convenient unitlessrepresentation equal to the ratio of absolute frequency, f_(A), and theNyquist frequency, f_(N), or ½ the sampling frequency 176, in (Equation3). Normalized frequency has a range in [−1, 1).

$\begin{matrix}{f = {\frac{f_{A}}{F_{N}} = \frac{2 \cdot f_{A}}{f_{S}}}} & \left( {{Equation}\mspace{14mu} 3} \right)\end{matrix}$3.0 The Analog Collection

As previously discussed, the Analog collection 114 consists of theSensor 158, the Filter 156 and the ADC 154 components. The Analogcollection 114 filters the analog three-phase voltage signal and theanalog three-phase current signal, ensuring that they meet physicalinterface, interference, noise, bandwidth, dynamic range and group delayrequirements established by the Digital collection 102.

The Sensor 158 represents the analog voltage and analog current sensors.The sensor reference, gain, frequency response and group delay may notbe entirely neglected, and can be included in any analysis. Further, thesensors 158 and the filters 156 can call for experimentalcharacterization when information is not available from themanufacturer.

The Filter 156 represents the analog signal paths including the voltagesensors and the current sensors, and anti-aliasing and gain stagefilters, which combine to meet ADC physical interface requirements, andpreserve dynamic range, bandwidth, and linearity of analog signals priorto quantization.

Aliasing occurs when analog signal content above the Nyquist frequencyis mapped to observable spectral content after quantization. The aliasedsignal content is only significant if it is observable, or of sufficientamplitude to effect a change in the discrete signal. Aliasing may not becompletely avoided, though it can be effectively eliminated byapplication of anti-aliasing filters. The anti-aliasing filters reducealiased spectral content to unobservable levels after sampling. Theanti-aliasing filters preferably demonstrate relatively constantmagnitude response, and linear phase, or constant group delay, over thepass band of operation.

The analog signal bandwidth, f_(B,A), is greater than or equal to 31times the rated fundamental frequency, in (Equation 4).ƒ_(N)>>ƒ_(B,A)>31·ƒ₀  (Equation 4)

The Filter 156 design constraints can be electively relaxed, inconsideration of available a priori knowledge. The supply voltage andstator current are inherently low bandwidth signals. Harmonicattenuation increases with frequency, due to motor geometry and design,and can provide at least 30 dB of attenuation at or above the Nyquistfrequency.

The Filter 156 includes a gain stage to ensure preservation of a dynamicrange of the analog signals, and to meet input signal interfacerequirements specified by the ADC 154. Transient periods of significantinrush occur immediately after the epoch of induction motor operation,and sustained operation above rated conditions is not uncommon.

A portion of the dynamic range of the system is reserved to accommodateobservation of at least 1.5 times the rated voltage 160 and the ratedcurrent 162 signal magnitudes, and discrimination of inrush conditions.The voltage and current signal gains are preferably independentlyspecified to a level where approximately ½ of the ADC input voltagerange is consumed during motor operation in rated conditions.

The Filter 156 architecture and order are not explicitly defined, andany suitable alternative designs which meet the Analog collection 114requirements are contemplated.

The ADC 154 converts the filtered analog voltage and current signalsinto quantized representations. Synchronous sampling can be assumed forconvenience if the inter-channel latency is negligible. Any significantlatency or asymmetric path delays are preferably reconciled in theSignal Extraction process 120, described in detail with respect to FIG.3.

Sampling frequency 176 is defined in terms of filter bandwidth, in(Equation 5). For example, a sampling frequency 176 of approximately 5kHz can be typical, in association with practical commercial filterdesigns.ƒ_(S)=2·ƒ_(N)>>2·ƒ_(B,A)  (Equation 5)

A dynamic range, D_(R), describes the magnitude of a minimallyobservable quantized signal, relative to the specific ADC input voltagerange, in units of effective bits or decibels, in (Equation 6).

The effective dynamic range of the Analog collection 114 is greater thanor equal to 72 dB, or approximately 12 bits. An Automatic Gain Control(AGC) can be desirable in some systems to preserve sufficient effectivedynamic range.D _(R)≧72dB  (Equation 6)

The ADC 154 produces the three-phase voltage 144, {right arrow over(v)}_(P,n), and the three-phase current 146, {right arrow over(i)}_(P,n).

4.0.0 The Digital Collection

The Digital collection 102 contains the data and process collections andinteractions to estimate slip 140 from the discrete three-phase voltage144 and the current 146 signals.

4.1.0 The Data Acquisition Process

Referring to FIG. 2, the Data Acquisition process 118 converts thethree-phase voltage 144 and the three-phase current 146 signals producedby the Analog collection 114 into complex representations 124 and 126,in a Complex Voltage process 202 and a Complex Current process 204.

4.1.1 The Complex Voltage Process

The Complex Voltage process 202, COMPLEX VOLTAGE, converts thethree-phase voltage 144 to an unbiased complex representation 124 as afunction of the rated voltage 160 and the neutral reference 170, in(Equation 7).v _(C,n)=_(COMPLEX VOLTAGE)({right arrow over (v)} _(P,n) ,v ₀ ,N_(R))  (Equation 7)v_(C,n) Complex Voltage 124.{right arrow over (v)}_(P,n) Three-Phase Voltage 144.v₀ Rated Voltage 160.N_(R) Neutral Reference 170.

The voltage signals are acquired in the form of sampled three-phasesequences, with one phase or dimension per source line. The complexrepresentations are synthesized through application of simple lineartransformations. Complex conversion is equivalent, reversible, lossless,mathematically convenient, and/or practically advantageous, relative tothree-phase representations.

Voltage definition is relative, and the three-phase voltage 144 isdescribed in terms of a line-neutral or a line-line source reference.The voltage sensor topologies are supported without preference, andsynthesis produces data with equivalent complex representations.Synthesis differences are limited to scale and rotation of the resultingcomplex data.

It is contemplated that any suitable matrices or vectors can be used toperform the linear transformations. For example, a complex synthesismatrix, {right arrow over (X)}_(C), is a constant matrix that can beused in the linear transformation which supports complex conversion, in(Equation 8).

$\begin{matrix}{{\overset{\rightarrow}{X}}_{C} = {\left( \frac{2}{3} \right) \cdot \begin{bmatrix}1.0 & {- 0.5} & {- 0.5} \\0.0 & {0.5 \cdot 3^{0.5}} & {{- 0.5} \cdot 3^{0.5}} \\1.0 & 1.0 & 1.0\end{bmatrix}}} & \left( {{Equation}\mspace{14mu} 8} \right)\end{matrix}$

Similarly, a complex synthesis vector, {right arrow over (u)}_(C), is aconstant vector that can be used to extract real and imaginarycomponents from an intermediate complex conversion vector, in (Equation9).{right arrow over (u)}_(C)=[1.0j0.0]  (Equation 9)

A biased complex voltage, v_(B,n), is synthesized through lineartransformation using the complex synthesis matrix and complex synthesisvector, as a function of neutral reference 170, N_(R), in (Equation 10).

The complex voltage 124 is normalized with respect to rated voltage 160,v₀.

Line-line voltage sensor topologies call for additional application of ascale factor and a coordinate rotation, achieved by multiplication witha complex constant.

$\begin{matrix}{v_{B,n} = \left\{ \begin{matrix}\begin{matrix}{{\frac{{\overset{\rightarrow}{u}}_{C}}{v_{0}} \cdot \left( {{\overset{\rightarrow}{X}}_{C} \cdot {\overset{\rightarrow}{v}}_{P,n}} \right) \cdot 3^{- 0.5} \cdot {\mathbb{e}}^{{- j}\frac{\pi}{6}}} =} \\{\frac{{\overset{\rightarrow}{u}}_{C}}{v_{0}} \cdot \left( {{\overset{\rightarrow}{X}}_{C} \cdot \begin{bmatrix}v_{P,{0 - 1},n} \\v_{P,{1 - 2},n} \\v_{P,{2 - 0},n}\end{bmatrix}} \right) \cdot 3^{- 0.5} \cdot {\mathbb{e}}^{{- j}\frac{\pi}{6}}}\end{matrix} & {\,_{NOT}\left( N_{R} \right)} \\{{\frac{{\overset{\rightarrow}{u}}_{C}}{v_{0}} \cdot \left( {{\overset{\rightarrow}{X}}_{C} \cdot {\overset{\rightarrow}{v}}_{P,n}} \right)} = {\frac{{\overset{\rightarrow}{u}}_{C}}{v_{0}} \cdot \left( {{\overset{\rightarrow}{X}}_{C} \cdot \begin{bmatrix}v_{P,0,n} \\v_{P,1,n} \\v_{P,2,n}\end{bmatrix}} \right)}} & N_{R}\end{matrix} \right.} & \left( {{Equation}\mspace{14mu} 10} \right)\end{matrix}$

It is implicit that the three-phase voltage samples are sequential inphase, signed, and approximately zero mean after quantization by the ADC154, and prior to complex conversion. Signed conversion consisting ofsimple subtraction can be utilized with some ADC architectures, thoughresidual bias present, due to imprecise ADC reference voltages, noise,or actual DC signal content, is preferably removed.

An exponential decay filter is a 1^(st) order Infinite Impulse Response(IIR) filter with coefficients directly specified from the bias filterbandwidth, f_(B,B), in (Equation 11). The IIR Filter architecture isdescribed in detail below with respect to FIG. 21.

$\begin{matrix}{f_{B,B} \approx \frac{0.01}{f_{N}}} & \left( {{Equation}\mspace{14mu} 11} \right)\end{matrix}$

A biased complex voltage mean, v_(B,n), is estimated by application ofan IIR filter, in (Equation 12).v _(B,n)=(1−ƒ_(B,B))· v _(B,n−1)+ƒ_(B,B) ·v _(B,n)=_(IIR)(v_(B,n),1−ƒ_(B,B),ƒ_(B,B))  (Equation 12)

The complex voltage 124, v_(C,n), is equal to the difference of thebiased complex voltage and its mean, in (Equation 13).v _(C,n) =v _(B,n) − v _(B,n)  (Equation 13)4.1.2 The Complex Current Process

The Complex Current process 204, _(COMPLEX CURRENT), converts thethree-phase current 146 to an unbiased complex representation 126 as afunction of rated current 162, in (Equation 14).i _(C,n)=_(COMPLEX CURRENT)({right arrow over (i)} _(P,n) ,i₀)  (Equation 14)i_(C,n) Complex Current 126.{right arrow over (i)}_(P,n) Three-Phase Current 146.i₀ Rated Current 162.

The current signals are acquired in the form of sampled three-phasesequences, with one phase or dimension per source line. The complexrepresentations 126 are synthesized through application of simple lineartransformations. Complex conversion can be equivalent, reversible,lossless, mathematically convenient, and/or practically advantageous,relative to three-phase representations.

Current definition is absolute, and complex current conversion isimplicitly defined in terms of a line-neutral source reference.

A biased complex current, i_(B,n), is synthesized through lineartransformation using the complex synthesis matrix, {right arrow over(X)}_(C), and complex synthesis vector, {right arrow over (u)}_(C), in(Equation 15).

The complex current 126 is normalized with respect to rated current 162,i₀.

$\begin{matrix}{i_{B,n} = {{\frac{{\overset{\rightarrow}{u}}_{C}}{i_{0}} \cdot \left( {{\overset{\rightarrow}{X}}_{C} \cdot {\overset{\rightarrow}{i}}_{P,n}} \right)} = {\frac{{\overset{\rightarrow}{u}}_{C}}{i_{0}}\left( {{\overset{\rightarrow}{X}}_{C} \cdot \begin{bmatrix}i_{P,0,n} \\i_{P,1,n} \\i_{P,2,n}\end{bmatrix}} \right)}}} & \left( {{Equation}\mspace{14mu} 15} \right)\end{matrix}$

It is implicit that three-phase current 146 samples are sequential inphase, signed, and approximately zero mean after quantization by the ADC154, and prior to complex conversion. Signed conversion consisting ofsimple subtraction can be utilized with some ADC architectures, thoughresidual bias present, due to imprecise ADC reference voltages, noise,or actual DC signal content, should be removed.

An exponential decay filter is a 1^(st) order IIR filter withcoefficients directly specified from the bias filter bandwidth, f_(B,B),in (Equation 16).

$\begin{matrix}{f_{B,B} \approx \frac{0.01}{f_{N}}} & \left( {{Equation}\mspace{14mu} 16} \right)\end{matrix}$

A biased complex current mean, i_(B,n), is estimated by application ofan IIR filter, in (Equation 17).ī _(B,n)=(1−ƒ_(B,B))·ī _(B,n−1)+ƒ_(B,B) ·i _(B,n)=_(IIR)(i_(B,n),1−ƒ_(B,B) ,f _(B,B))  (Equation 17)

The complex current 126, i_(C,n), is equal to the difference of thebiased complex current and its mean, in (Equation 18).i _(C,n) =i _(B,n) −ī _(B,n)  (Equation 18)4.2.0 The Signal Extraction Process

The Signal Extraction process 120 filters the complex voltage signal 124and the complex current signal 126 and produces a data collectionincluding a complex fundamental voltage 128, a complex fundamentalcurrent 130 and a complex residual current 132, a complex input power134, an instantaneous fundamental frequency 136, and an approximate slip138.

Referring to FIG. 3, the Signal Extraction process 120 consists of aComplex Voltage Components process 302, a Complex Current Componentsprocess 304, a Latency Compensation process 306, a Complex Input Powerprocess 308, a Fundamental Frequency process 310 and an Approximate Slipprocess 312.

4.2.1 The Complex Voltage Components Process

The Complex Voltage Components process 302,_(COMPLEX VOLTAGE COMPONENTS), extracts an estimate of the complexfundamental voltage 128 as a function of the complex voltage 124, therated fundamental frequency 166 and the sampling frequency 176, in(Equation 19).v _(F,n)=_(COMPLEX VOLTAGE COMPONENTS)(v _(C,n),ƒ₀,ƒ_(S))  (Equation 19)v_(F,n) Complex Fundamental Voltage 128.v_(C,n) Complex Voltage 124.ƒ₀ Rated Fundamental Frequency 166.ƒ_(S) Sampling Frequency 176.

The complex fundamental voltage 128 is a complex exponential componentof the complex voltage 124 proximate to the normalized rated fundamentalfrequency. The complex fundamental voltage 128 can be estimated by anysuitable filter such as, for example, through application of a staticband pass filter or a Complex Single Frequency (CSF) adaptive filter.

CSF filters have inherent superior performance, relative to static bandpass filter topologies, due to the ability to dynamically predict ormatch the magnitude and phase of a complex exponential component ofinterest in a primary signal. CSF filters are computationally simple,and flexible, as they are readily tunable to any frequency of interest.

CSF filters consist of a Voltage Controlled Oscillator (VCO) and a meansof complex coefficient adaptation, which are combined to supportsynthesis of complex incident, reference and error signals, with respectto an external complex primary signal. The CSF filter is described indetail below with respect to FIG. 23. In a quasi-stationary environment,relative to the response of the filter, an optimum complex coefficientvalue can be found to minimize the resulting error signal, in a leastsquares sense, resulting in synthesis of a reference signal thatapproximates a component of interest in the primary signal.

Convergence, the time required to find the optimum complex coefficient,is inversely proportional to the coefficient adaptation rate 412, μw.Misadjustment, the estimation noise introduced by the adaptive process,is proportional to the coefficient adaptation rate 412. Fasterconvergence results in increased estimation noise. The nominalcoefficient adaptation rate 412 is 1.0e−3.

Momentum is a nonlinear technique applied to improve convergence time,or the effort expended to find the optimum complex coefficient value,with potential implications on stability and misadjustment. Coefficientmomentum 414, α_(W), accelerates complex coefficient change along aconsistent trajectory. The nominal coefficient momentum 414 is zero.

The complex fundamental voltage 128, v_(F,n), is extracted from a CSFfilter with the complex voltage 124 assigned to the complex primarysignal 2304, and a synthesis frequency 2302 equal to the normalizedrated fundamental frequency 166, in (Equation 20).

$\begin{matrix}{v_{F,n} =_{CSF}\left( {v_{C,n},\frac{f_{0}}{f_{N}},\mu_{W},\alpha_{W}} \right)} & \left( {{Equation}\mspace{14mu} 20} \right)\end{matrix}$

The complex fundamental voltage 128 is assigned from the complexreference signal 2308, and the complex error signal 2306 is notretained.

4.2.2 The Complex Current Components Process

The Complex Current Components process 304,_(COMPLEX CURRENT COMPONENTS), extracts estimates of the complexfundamental current 130 and the complex residual current 132 as afunction of the complex current 126, the rated fundamental frequency 166and the sampling frequency 176, in (Equation 21).⊙i _(F,n) ,i _(R,n)┘=_(COMPLEX CURRENT COMPONENTS)(i_(C,n),ƒ₀,ƒ_(S))  (Equation 21)i_(F,n) Complex Fundamental Current 130.i_(R,n) Complex Residual Current 132.i_(C,n) Complex Current 126.ƒ₀ Rated Fundamental Frequency 166.ƒ_(S) Sampling Frequency 176.

The complex fundamental current 130, a complex exponential component ofcomplex current 126 proximate to the normalized rated fundamentalfrequency, is estimated by any suitable means such as, for example,through application of static band pass filters or a CSF adaptivefilter. The complex residual current 132 is the remainder, or differenceof the complex current 126 and the complex fundamental current 130.

The complex fundamental current 130, i_(F,n), and the complex residualcurrent 132, i_(R,n), are extracted from the CSF filter with complexcurrent 126 assigned to the complex primary signal 2304, and synthesisfrequency 2302 equal to the normalized rated fundamental frequency, in(Equation 22). As described with respect to the Complex VoltageComponents process 302, the nominal coefficient adaptation rate 412,μ_(W), is 1.0e−3 and the nominal coefficient momentum 414, α_(W), iszero.

$\begin{matrix}{\left\lbrack {i_{F,n},i_{R,n}} \right\rbrack =_{CSF}\left( {i_{C,n},\frac{f_{0}}{f_{N}},\mu_{W},\alpha_{W}} \right)} & \left( {{Equation}\mspace{14mu} 22} \right)\end{matrix}$

Complex fundamental current 130 is assigned from the complex referencesignal 2308, and complex residual current 132 is assigned from thecomplex error signal 2306.

4.2.3 The Latency Compensation Process

Referring to FIG. 4, the Latency Compensation process 306,_(LATENCY COMPENSATION), reconciles the latencies associated with thecomplex fundamental voltage 128 and the complex fundamental current 130signals, ensuring temporal alignment among all signals on whichsubsequent multiple dependencies exist, in (Equation 23).└v _(F,n) ,i _(F,n)┘=_(LATENCY COMPENSATION)(v _(F,n) ,i_(F,n),ƒ_(S))  (Equation 23)v_(F,n) Complex Fundamental Voltage 128.i_(F,n) Complex Fundamental Current 130.v_(F,n) Complex Fundamental Voltage 128.i_(F,n) Complex Fundamental Current 130.ƒ_(S) Sampling Frequency 176.

The complex fundamental voltage 128 and the complex fundamental current130 are delayed to support temporal synchronization with other signals.The input and output symbols for these signals remain unchanged fornotational convenience, though subsequent references imply the latencycompensated versions.

Independent processing paths incur an associated penalty, or latency,due to the specific set of architectures and methods which define apath. Compound signals are defined in terms of multiple signalsextracted from different dependent processing paths. The latency of eachdependent processing path is reconciled such that the effective latencyof each dependent signal is equal. The Latency Compensation process 306aligns dependent signals in time prior to evaluation of a compoundsignal.

The Latency Compensation process 306 can be accomplished by evaluatingthe aggregate delays associated with various dependent signal processingpaths, identifying the slowest, or longest latency path, and definingappropriate additional latencies required for each of the remainingdependent paths so that the latencies are equal. The LatencyCompensation process 306 appends a specific causal delay to eachdependent signal processing path as desired, resulting in an equivalenteffective latency for all dependent paths.

The Latency Compensation process 306 is clarified by analysis of therelative inter-path latencies resulting from estimation of thefundamental frequency 136, the complex fundamental voltage 128, thecomplex fundamental current 130, and a saliency frequency 426 (discussedbelow with respect to FIG. 8), as a function of the complex voltage 124and the complex current 126.

Though fundamental frequency 136 and saliency frequency 426 have not yetbeen formally presented, it is sufficient to limit the currentdescription to the definition of signal dependencies and relativelatencies, so that the requirements and means for latency compensationcan be clearly understood.

The timing diagram of FIG. 4 illustrates the production of varioussignals, and their associated latencies, in the framework of a temporalmap. The abscissa is in units of samples, with an arbitrary epoch atsample n, and a range of n_(L) samples. The range corresponds to theworst-case latency in any processing path.

The processing paths are identified with bold directional lines betweenenumerated variables. The processing paths traverse vertically,indicating that negligible latency is incurred in production of thevariable, or horizontally, indicating significant latency.

The architectures associated with processing paths are specified inclosed braces. Dependencies which effect latency, including the samplingfrequency 176, adaptive parameters, and filter coefficients 402 and 404,are associated with specific architectures by light vertical directionallines. The adaptive parameters include, for example, a frequencyestimation 410, the coefficient adaptation rate 412, the coefficientmomentum 414, a filter bandwidth 416, a frequency adaptation rate 418,and a frequency momentum 420. The filter coefficients include, forexample, saliency filter coefficients 404 and fundamental filtercoefficients 402.

Processing paths originate with the complex voltage 124 and the complexcurrent 126, and terminate with the fundamental frequency 136, thecomplex fundamental voltage 128, the complex fundamental current 130,and the saliency frequency 426. The epoch and terminus signals areenclosed in circles.

The intermediate signals embedded in the timing diagram of FIG. 4 neednot be delayed and reconciled with terminus signals. The latency shouldbe reconciled when two or more independent signal paths are combined toproduce a new signal. Through judicious application of the LatencyCompensation process 306 just-in-time, we can limit the scope of thecompensation to adding independent delays to the complex fundamentalvoltage 128 and complex fundamental current 130 signal paths. Noadditional latency compensation is needed.

Four processing paths are defined, specifying the production of thefundamental frequency 136, the complex fundamental voltage 128, thecomplex fundamental current 130, and the saliency frequency 426. Thefundamental frequency 136 and the saliency frequency 426 processingpaths can be defined to be equivalent with respect to latency. Thecomplex fundamental voltage 128 and the complex fundamental current 130processing paths are synchronized to frequency estimation processingpaths by latency compensation, consisting of independent static delays.

The fundamental frequency 136, f_(0,n), is estimated from the complexvoltage 124. According to some aspects, the complex voltage 124 producesa complex baseband voltage 422 by applying a VCO and a FIR or an IIRfilter (described in detail with respect to FIG. 21). The VCO latency isnegligible, though the filter latency can be significant. One of severalalternative frequency estimation architectures can be applied to extractthe fundamental frequency 136 from the complex baseband voltage 422.

The FIR or IIR Filter latency is dependent on the fundamental filtercoefficients 402 and the sampling frequency 176, and represents groupdelay. The frequency estimation latency is highly dependent on thespecific architecture, the adaptive coefficients and the samplingfrequency 176. Alternative methods of iterative frequency estimationinclude, for example, Direct, Phase Discriminator (PD), and Phase LockedLoop (PLL) analysis. The PD and the PLL architectures are describedbelow with respect to FIG. 24 and FIG. 25, respectively.

The complex fundamental voltage 128, v_(F,n), is estimated from thecomplex voltage 124. A CSF filter can be used to extract the complexfundamental voltage 128 from complex voltage 124. The CSF filter latencyis negligible, neglecting convergence and bandwidth limitations, as itis an adaptive predictor. A causal integer delay, n_(L), produces thelatency compensated complex fundamental voltage 128 from theuncompensated signal.

The complex fundamental current 130, i_(F,n), is estimated from thecomplex current 126. A CSF filter can be used to extract the complexfundamental current 130 and the complex residual current 132, from thecomplex current 126. Again, the CSF filter latency is negligible,neglecting convergence and bandwidth limitations, as it is an adaptivepredictor. A causal integer delay, n_(L), produces the latencycompensated complex fundamental current 130 from the uncompensatedsignal.

The saliency frequency 426, f_(H,n), is estimated from the complexcurrent 126. According to some aspects, a CSF filter is used to extractthe complex fundamental current 130 and the complex residual current132, from the complex current 126. The CSF filter latency is negligible,neglecting convergence and bandwidth limitations, as it is an adaptivepredictor. The complex residual current 132 produces the complexbaseband current 424 by applying a VCO and FIR or IIR filter. The VCOlatency is negligible, though filter latency is significant. One ofseveral alternative frequency estimation architectures can be applied toextract saliency frequency from complex baseband current 424.

The FIR or IIR Filter latency is dependent on the saliency filtercoefficients 404 and the sampling frequency 176, and represents groupdelay. Frequency estimation latency is highly dependent on specificarchitecture, adaptive coefficients and sampling frequency. Alternativemethods of iterative frequency estimation include, for example, Direct,PD, and PLL analysis.

To reduce redundant efforts to define the specific causal integer delayto facilitate latency compensation 306 in similar systems, latencycoefficients are introduced to support delay definition as a function ofthe frequency estimation architecture and the sampling frequency 176.

The latency compensation coefficients provided implicitly depend oncertain assumptions regarding the design of a specific system. CSFfilter adaptive parameters for the Complex Voltage Component 302 and theComplex Current Component 304 extraction should be identical, and equalto the default parameters that will be described below with respect toFIG. 23. The fundamental frequency 136 and the saliency frequency 426estimation architectures, constants, and adaptive parameters should beidentical, and equal to the default PD or PLL parameters that will bedescribed with respect to FIG. 24 and FIG. 25, respectively. Thefundamental filter coefficients 402 and the saliency filter coefficients404 should specify FIR filters with filter order, M, equal to ¼ thesampling frequency 176.

It is contemplated that alternative IIR filter architectures oralternative FIR filter orders can be utilized to accomplish the latencycompensation process 306. Accordingly, the latency compensation delayspecified in the following description can be modified to account forthe change in filter group delay resulting from these alternativeembodiments.

A designer is provided with a number of options to support flexibledesign in a myriad of environments with differing processor or memoryresources. Independent of flexibility in the algorithm, resultingdesigns and implementations are likely to demonstrate considerablediversity. Therefore, it is desirable to verify that the latencycompensation for a specific implementation is correct.

The latency coefficients, {right arrow over (c)}_(L,m), are specifiedfrom analysis of supported fundamental frequency 136 and saliencyharmonic frequency estimation 426 methods, in (Equation 24).

$\begin{matrix}{{\overset{\rightarrow}{c}}_{L,m} = \left\{ \begin{matrix}\left\lbrack {{- 163}\mspace{14mu} 0.0850} \right\rbrack & {m_{F} =_{DIRECT}} \\\left\lbrack {60\mspace{45mu} 0.0872} \right\rbrack & {m_{F} =_{PD}} \\\left\lbrack {190\mspace{31mu} 0.0864} \right\rbrack & {m_{F} =_{PLL}}\end{matrix} \right.} & \left( {{Equation}\mspace{14mu} 24} \right)\end{matrix}$

The latency coefficients describe a 1^(st) order polynomial and aredefined in ascending order, with respect to spatial index _(m). The zeroorder coefficient, c_(L,0), expresses a constant latency offset. Thefirst order coefficient, c_(L,1), is a function of the samplingfrequency 176, and expresses a dynamic latency rate.

A latency compensation delay, n_(L), is the delay, rounded to thenearest integer, expressed as a polynomial product of the latencycoefficients and the sampling frequency 176, in units of samples, in(Equation 25).n _(L) =└c _(L,0) +c _(L,1)·ƒ_(S)┘  (Equation 25)

The complex fundamental voltage 128, v_(F,n), is delayed by the latencycompensation delay samples, in (Equation 26).v _(F,n) =v _(F,n-n) _(L) =_(DELAY)(v _(F,n) ,n _(L))  (Equation 26)

The complex fundamental current 130, i_(F,n), is delayed by latencycompensation samples, in (Equation 27).i _(F,n) =i _(F,n-n) _(L) =_(DELAY)(i _(F,n) ,n _(L))  (Equation 27)

The complex fundamental voltage 128 and the complex fundamental current130 are delayed to support any temporal synchronization with othersignals. The input and output symbols for these signals remain unchangedfor notational convenience, though subsequent references imply thelatency compensated versions.

4.2.4 The Complex Input Power Process

The Complex Input Power process 308, _(COMPLEX INPUT POWER), extracts anestimate of the complex input power 134 as a function of the complexfundamental voltage 128, the complex fundamental current 130 and therated power factor 168, in (Equation 28).p _(F,n)=_(COMPLEX INPUT POWER)(v _(F,n) ,i _(F,n) ,P _(F))  (Equation28)p_(F,n) Complex Input power 134.v_(F,n) Complex Fundamental Voltage 128.i_(F,n) Complex Fundamental Current 130.P_(F) Rated Power Factor 168.

The complex input power 134, p_(F,n), is normalized to rated inputpower, as the scaled product of the complex fundamental voltage 128 andthe conjugate of the complex fundamental current 130, in (Equation 29).

$\begin{matrix}{p_{F,n} = \frac{3^{0.5} \cdot v_{F,n} \cdot i_{F,n}^{*}}{2 \cdot P_{F}}} & \left( {{Equation}\mspace{14mu} 29} \right)\end{matrix}$

Referring to FIG. 5, the complex input power 134 estimation isillustrated for a 15 HP, 6 pole motor 150 operating in rated voltage 160of 460 V and dynamic current conditions of 10.0-18.5 A, for a period of60.0 seconds, at a sampling frequency 176 of 10 kHz. The normalized realinput power is identified as 502 and the imaginary input power isidentified as 504. The motor 150 was operated in near rated conditions,or approximately unity normalized real input power, at the epoch of thesignal, and the load 152 was gradually decreased and increased totraverse the observable below rated range of operation.

4.2.5 The Fundamental Frequency Process

Referring to FIG. 6, the Fundamental Frequency process 310,_(FUNDAMENTAL FREQUENCY), extracts an instantaneous estimate offundamental frequency 136 as a function of the complex voltage 124, therated fundamental frequency 166 and the sampling frequency 176, in(Equation 30).ƒ_(0,n)=_(FUNDAMENTAL FREQUENCY)(v _(C,n),ƒ₀,ƒ_(S))  (Equation 30)ƒ_(0,n) Fundamental Frequency 136.v_(C,n) Complex Voltage 124.ƒ₀ Rated Fundamental Frequency.ƒ_(S) Sampling Frequency.

The fundamental frequency 136 is dynamic, due to variation in the supplyvoltage 148 and the load 152. Though a nominal rated fundamentalfrequency 166 is available, a more accurate instantaneous fundamentalfrequency 136 estimate can be obtained. The complex voltage 124 isselected as a preferred source for analysis, as it has a relatively highsignal to noise ratio, in terms of the complex fundamental voltage 128and the complex residual voltage components; however, it is contemplatedthat any other suitable source can be selected to obtain an estimate ofthe fundamental frequency 136.

A demodulation process applies a VCO 602 to mix the rated fundamentalfrequency 166 in the complex voltage 124 to complex baseband, or a zeronominal frequency. An FIR or IIR filter 604 can be applied to band limitthe mixed voltage 610, producing a complex baseband voltage 422. Tocomplete the demodulation process, a residual frequency contained in thecomplex baseband voltage 422 is extracted, resulting in an accuratefundamental frequency 136 estimate.

According to alternative aspects, the fundamental frequency 136 can beestimated by any application of frequency domain or time domain methods,employing static or adaptive architectures, in parametric ornon-parametric form, producing instantaneous or aggregate frequencyestimates. For example, frequency domain methods of frequency estimationinclude the Fourier transform, Fast Fourier Transform (FFT), PowerSpectral Density (PSD), Chirp Z Transform (CZT), Autoregressive (AR)model, Moving Average (MA) model, Autoregressive Moving Average (ARMA)model, Prony method, Pisarenko Harmonic Decomposition (PHD), MaximumLikelihood Method (MLM), Least Squares Spectral Analysis (LSSA), theGoertzel method, and the like. Non-limiting examples of time domainmethods of frequency estimation include the Phase Locked Loop (PLL),Phase Discriminator (PD), discrete phase differentiation, and variousmethods of Frequency Modulation (FM) signal demodulation.

The frequency and time domain methods of frequency estimation can beinstantaneous, representing transient dynamics in a source signal, oraggregate, representing a quasi-stationary weighted response over afinite period of observation. Various methods can appropriately beapplied to produce iterative transient frequency estimations, whileothers are exclusively suitable for operation on a signal defined over afinite period of observation. The enumerated frequency estimationmethods described with respect to any frequency quantity (e.g.,fundamental frequency 136, saliency frequency 426, etc.) are inclusiveand representative, not exclusive.

4.2.5.1 Obtaining The Complex Baseband Voltage

A complex incident signal 608, X_(D,n), is synthesized by the VCO 602 atthe normalized rated fundamental frequency, f₀, in (Equation 31).

$\begin{matrix}{x_{D,n} =_{VCO}\left( \frac{f_{0}}{f_{N}} \right)} & \left( {{Equation}\mspace{14mu} 31} \right)\end{matrix}$

A complex mixed voltage 610, V_(D,n), is formed as the product of theconjugate of the complex incident signal 608 and the complex voltage124, in (Equation 32).v _(D,n) =v _(C,n) ·x* _(D,n)  (Equation 32)

The fundamental filter coefficients 402, {right arrow over (a)}_(U) and{right arrow over (b)}_(U), are statically designed with a fundamentalfilter bandwidth, f_(B,U), defined in terms of the normalized ratedfundamental frequency, in (Equation 33).

$\begin{matrix}{f_{B,U} = {0.1 \cdot \frac{f_{0}}{f_{N}}}} & \left( {{Equation}\mspace{14mu} 33} \right)\end{matrix}$

Suitable filters 604 include, for example, linear phase filters such asvarious FIR designs with memory depth equal to approximately ¼ second,and IIR Bessel filters with comparable performance. The filterarchitecture selection is dependent largely upon the computational anddesign complexity, and numerical stability.

A static filter design is practical, due to the small finite set ofrated fundamental frequencies supported. The appropriate fundamentalfilter coefficients 402 can be selected from a predefined set andapplied in a deterministic manner.

The fundamental filter coefficients 402 are applied to filter thecomplex mixed voltage 610, producing complex baseband voltage 422,v_(U,n), in (Equation 34).v _(U,n)=_(IIR)(v _(D,n) ,{right arrow over (a)} _(U) ,{right arrow over(b)} _(U))  (Equation 34)4.2.5.2 Estimating The Fundamental Frequency

A demodulation process removes the rated fundamental frequency 166, orthe nominal fundamental frequency expected during motor operation inrated conditions, by mixing the complex voltage 124 by the conjugate ofthe complex exponential at the normalized rated fundamental frequency.The complex mixed voltage 610 can be band limited through application ofa fundamental filter to isolate the remaining residual fundamentalharmonic from interference sources, producing the complex basebandvoltage 422. The demodulation process is completed by estimating theresidual fundamental frequency in the complex baseband voltage 422. Thefundamental frequency 136 is the sum of the normalized rated fundamentalfrequency and the residual fundamental frequency.

The fundamental frequency estimation methods are analogous todemodulation of an FM signal. The complex baseband voltage 422 isextracted through a process of carrier removal. The residual frequencyestimation extracts the instantaneous frequency of the fundamentalharmonic, relative to the normalized rated fundamental frequency, orcarrier. Fundamental frequency 136 is expressed as the sum of thecarrier frequency and residual frequency.

Alternative methods such as, for example, Direct, PD, and PLL methods410, m_(F), of estimating fundamental frequency 136 provide theflexibility to increase precision at the expense of computationalcomplexity and frequency tracking rate, expanding the practicalapplicability of the solution.

4.2.5.2.1 The Direct Method of Estimating Fundamental Frequency

The direct estimation of residual fundamental frequency is the discretederivative of the complex baseband voltage phase.

A residual fundamental phase, φ_(U,n), is the normalized phase of thecomplex baseband voltage 422, estimated through application of acontiguous arctangent function, in (Equation 35).

$\begin{matrix}{\varphi_{U,n} =_{{TAN}^{- 1}}{{\left( \frac{\,_{IMAG}\left( v_{U,n} \right)}{\,_{REAL}\left( v_{U,n} \right)} \right) \cdot \frac{1}{\pi}} + {0.5 \cdot \left( {1 -_{SIGN}\left( {}_{REAL}\left( v_{U,n} \right) \right)} \right) \cdot_{SIGN}\left( {}_{IMAG}\left( v_{U,n} \right) \right)}}} & \left( {{Equation}\mspace{14mu} 35} \right)\end{matrix}$

A normalized phase is extracted by an inverse tangent, _(TAN) ⁻¹,applied to the ratio of imaginary and real complex components, scaled bythe inverse of π to normalize the result, and adjusted to reconcile thequadrant of operation. An arctangent method can be practically definedin terms of polynomial approximation, indexed table, or some combinedmethod.

A residual fundamental frequency, f_(R,n), is the discrete derivative ofthe residual fundamental phase, bounded by a practical range, in(Equation 36).

$\begin{matrix}{f_{R,n} = \left. {{\frac{\mathbb{d}\;}{\mathbb{d}n}\left( \varphi_{U,n} \right)} \approx {\,_{{MA}\; X}\left( {}_{MIN}{\left( {{\varphi_{U,n} - \varphi_{U,{n - 1}}},f_{R,M}} \right),{- f_{R,M}}} \right)}} \right|_{f_{R,M} = {{5.0e} - {3 \cdot \frac{f_{0}}{f_{N}}}}}} & \left( {{Equation}\mspace{14mu} 36} \right)\end{matrix}$

The fundamental frequency 136, f_(0,n), is the sum of the normalizedrated fundamental frequency and the residual fundamental frequency, in(Equation 37).

$\begin{matrix}{f_{0,n} = \left. {{\frac{f_{0}}{f_{N}} + f_{R,n}} \approx {\frac{f_{0}}{f_{N}} + {\,_{{MA}\; X}\left( {}_{MIN}{\left( {{\varphi_{U,n} - \varphi_{U,{n - 1}}},f_{R,M}} \right),{- f_{R,M}}} \right)}}} \right|_{f_{R,M} = {{5.0e} - {3 \cdot \frac{f_{0}}{f_{N}}}}}} & \left( {{Equation}\mspace{14mu} 37} \right)\end{matrix}$

The Direct method contiguous phase estimation and discretedifferentiation are computationally simple, with negligible associatedlatency, and no introduced limitation in the frequency tracking rate, orreduction in signal bandwidth. The frequency tracking rate is the rate,in units of normalized frequency per second, at which the fundamentalfrequency 136 changes, a function of supply 148 and aggregate load 152dynamics.

The Direct method memory depth is unity, resulting in faster tracking atthe expense of increased estimation noise. Latency represents thecumulative computational delay, which is reconciled by shifting resultsto align them temporally with the source signals employed to producethem.

The Direct method frequency estimates are equivalent to thesuperposition of independent estimates of all complex baseband voltage422 frequency components, over the fundamental filter bandwidth. TheDirect method forms an aggregate and biased estimate of the fundamentalfrequency 136. Practically, the fundamental filter bandwidth can containsignificant interference which can result in unacceptable performance inmany environments.

4.2.5.2.2 The PD Method of Estimating Fundamental Frequency

A PD is an adaptive filter which estimates the residual frequency of thecomplex baseband voltage 422 through a process of input normalization,and adaptation of a complex coefficient which reconciles the phasedifference between sequential normalized samples, encoding theinstantaneous frequency in the phase of the complex coefficient. PDarchitecture is described in detail with respect to FIG. 24.

The complex error is minimized when the complex coefficient rotates thedelayed complex incident signal in phase to compensate for the phasedifference between sequential normalized samples. Frequency is definedas phase difference with respect to time.

The instantaneous frequency of the complex baseband voltage 422 isencoded in the phase of the complex coefficient. No capability exists todiscriminate on the basis of frequency between complex exponentialcomponents in the complex baseband voltage 422. An aggregateinstantaneous frequency estimate is extracted from the superposition ofcomponents present in the signal.

The PD method offers improved accuracy, relative to the Direct method,due to the noise reduction inherent in the memory depth associated withthe adaptive complex coefficient. However, the improved accuracy comesat a cost of a modest increase in latency, and a minimal reduction inthe frequency tracking rate. Latency represents the cumulativecomputational delay, which is reconciled by shifting results to alignthem temporally with the source signals employed to produce them.

PD convergence, misadjustment, frequency tracking rate, and bandwidthare dependent upon the selection of the adaptive parameters, which maybe judiciously defined to optimally support a specific environment. Anominal coefficient adaptation rate 412 and a nominal coefficientmomentum 414 are 2.0e−3 and 1.5e−1, respectively.

The residual fundamental frequency, f_(R,n), is estimated from theapplication of a PD to the complex baseband voltage 422, bounded by apractical range, in (Equation 38).

$\begin{matrix}{f_{R,n} = \left. {\,_{{MA}\; X}\left( {}_{MIN}{\left( {}_{PD}{\left( {v_{U,n},\mu_{W},\alpha_{W}} \right),f_{R,M}} \right),{- f_{R,M}}} \right)} \right|_{f_{R,M} = {{5.0e} - {3 \cdot \frac{f_{0}}{f_{N}}}}}} & \left( {{Equation}\mspace{14mu} 38} \right)\end{matrix}$

The fundamental frequency 136, f_(0,n), is the sum of normalized ratedfundamental frequency and the residual fundamental frequency, in(Equation 39).

$\begin{matrix}{f_{0,n} = {{\frac{f_{0}}{f_{N}} + f_{R,n}} = \left. {\frac{f_{0}}{f_{N}} +_{MAX}\left( {}_{MIN}{\left( {}_{PD}{\left( {v_{U,n},\mu_{W},\alpha_{W}} \right),f_{R,M}} \right),{- f_{R,M}}} \right)} \right|_{f_{R,M} = {{5.0e} - {3 \cdot \frac{f_{0}}{f_{N}}}}}}} & \left( {{Equation}\mspace{14mu} 39} \right)\end{matrix}$4.2.5.2.3 The PLL Method of Estimating Fundamental Frequency

A Phase Locked Loop (PLL) is a closed loop adaptive filter optimallysuited to accurately estimate an instantaneous frequency in a dynamicenvironment with significant in-band interference. Adaptive frequencysynthesis and interference rejection support the identification andtracking of a complex exponential component of interest in a complexprimary signal. PLL architecture is described in detail below withrespect to FIG. 25.

A PLL consists of a VCO 602, a mixer, an IIR filter 2100, a PD 2400, anda means of frequency adaptation 2508. The VCO 602 synthesizes a complexexponential signal at an instantaneous frequency of interest. Theproduct of the conjugate of the complex exponential signal and thecomplex primary signal is band limited with the IIR filter, resulting ina complex baseband signal 2502. The complex baseband is a convenientrepresentation of a complex signal with zero nominal frequency. Theresidual frequency, or estimation error, of the complex baseband signalis estimated by the PD 2400, and employed in adaptation of the synthesisfrequency 2506. The residual fundamental frequency estimated by the PD2400 is used to iteratively adapt the VCO synthesis frequency 2506,forcing the complex exponential signal to remain at a nominal zerofrequency, centered in the complex baseband.

The PLL method offers improved accuracy, relative to the Direct methodand the PD method, due to the noise reduction inherent in the memorydepth of the integrated PD and iterative adaptation of the synthesisfrequency, and the optional application of an IIR filter to reducein-band interference. However, the improved accuracy comes at a furthercost of a minimal increase in latency, and a minimal reduction in thefrequency tracking rate, relative to the PD method.

A PLL filter bandwidth 2504, f_(B,P), can be defined according to thenature of the complex baseband voltage 422 environment. The bandwidthcan be increased, in return for significant reduction in latency andimproved frequency tracking rate, at the cost of increased aggregatefrequency estimation error. Unity bandwidth selection, which can beappropriate in environments with limited in-band interference,effectively excises the PLL IIR filter 2100, eliminating latencycontributions by the filter and maximizing frequency tracking rate.Nominal PLL filter bandwidth 2504 is 1.0.

The PLL convergence, misadjustment, frequency tracking rate, andbandwidth are dependent upon selection of the adaptive parameters, whichcan be judiciously defined to optimally support a specific environment.The nominal coefficient adaptation rate 412 and the nominal coefficientmomentum 414 are 5.0e−3 and zero, respectively. The nominal frequencyadaptation rate 418 and the nominal frequency momentum 420 are 2.0e−3and 3.5e−1, respectively.

The residual fundamental frequency, f_(R,n), is estimated from theapplication of a PLL to the complex baseband voltage 422, initialized tothe expected residual fundamental frequency and selected PLL filterbandwidth, bounded by a practical range, in (Equation 40).

$\begin{matrix}{f_{R,n} = \left. {\,_{{MA}\; X}\left( {}_{\;_{MIN}}{\left( {}_{PLL}{\left( {v_{U,n},f_{R,0},f_{B,H},\mu_{W},\alpha_{W},\mu_{F},\alpha_{F}} \right),f_{R,M}} \right),{- f_{R,M}}} \right)} \right|_{f_{R,M} = {{5.0e} - {3 \cdot \frac{f_{0}}{f_{N}}}}}} & \left( {{Equation}\mspace{14mu} 40} \right)\end{matrix}$

An initial PLL frequency, f_(R,0), is the expected residual fundamentalfrequency, or zero, as the complex incident signal is synthesized at thenormalized rated fundamental frequency, in (Equation 41).ƒ_(R,0)=0  (Equation 41)

The fundamental frequency 136, f_(0,n), is the sum of the normalizedrated fundamental frequency and the residual fundamental frequency, in(Equation 42).

$\begin{matrix}{f_{0,n} = {{f_{D} + f_{R,n}} = \left. {f_{D} +_{MAX}\left( {}_{\;_{MIN}}{\left( {}_{PLL}{\left( {v_{U,n},f_{R,0},f_{B,P},\mu_{W},\alpha_{W},\mu_{F},\alpha_{F}} \right),f_{R,M}} \right),{- f_{R,M}}} \right)} \right|_{f_{R,M} = {{5.0e} - {3 \cdot \frac{f_{0}}{f_{N}}}}}}} & \left( {{Equation}\mspace{14mu} 42} \right)\end{matrix}$

The process of fundamental frequency estimation is continuous anditerative over a period of uninterrupted motor operation. Initializationand convergence occurs subsequent to each motor start.

Referring to FIG. 7, a normalized fundamental frequency estimationemploying the PLL method is illustrated, for a 15 HP, 6 pole motoroperating in near rated conditions, for a period of 60.0 seconds, at asampling frequency of 10 kHz.

The frequency and amplitude of variations observable in the fundamentalfrequency 136 can be considered typical for a mains supply. The range ofnormalized frequency variation over the period of observation isapproximately 3.3e−6, corresponding to an absolute frequency range of0.0165 Hz, relative to the rated fundamental frequency 166 of 60.0 Hz.

4.2.6 The Approximate Slip Process

The Approximate Slip process 312, _(APPROXIMATE SLIP), extracts anapproximation of the approximate slip 138 as a function of the complexinput power 134, the rated speed 164 and the rotor temperature, in(Equation 43).s _(P,n)=_(APPROXIMATE SLIP)(p _(F,n) ,r ₀ ,P,Θ _(TR,n))  (Equation 43)s_(P,n) Approximate Slip 138.p_(F,n) Complex Input Power 134.r₀ Rated Speed 164.Θ_(TR,n) Rotor Temperature. (Θ_(TR,0)).

An approximate slip 138 is the product of a rated slip and thenormalized real input power. Rotor temperature compensation can beemployed to improve approximate slip estimation 312, if a reasonablyaccurate rotor temperature estimate is available. Approximate slip 138is not as accurate as slip estimation based on harmonic analysis(discussed below with respect to FIGS. 8-9 and 13), though it is simpleand independent of rotor slot quantity 142.

The rated slip, s_(0,n), is the nominal slip expected during motoroperation in rated voltage and rated current conditions, in (Equation44).

$\begin{matrix}{s_{0,n} = {{1 - \frac{r_{0} \cdot P}{120 \cdot f_{0,n} \cdot f_{N}}} \approx {1 - \frac{r_{0} \cdot P}{120 \cdot f_{0}}}}} & \left( {{Equation}\mspace{14mu} 44} \right)\end{matrix}$

Slip demonstrates a temperature dependence which is generally notnegligible, and it is implicit that the rated operation is definedcorresponding to a specific rotor temperature. The precise rotortemperature associated with rated operation is typically not specifiedby the manufacturer, though experimental analysis reveals anapproximately linear relationship between slip and rotor temperature.

A plurality of temperature coefficients, c_(T,m,n), defining rotortemperature compensation, are defined from analysis of data from severalrepresentative motors 150 in various thermal conditions, in (Equation45).c _(T,m,n)=└−1.95e−3_(MAX)(_(MIN)(2.75e−3·Θ_(TR,n)+0.805,1.0),0.805)┘  (Equation 45)

The temperature coefficients describe a 1^(st) order polynomial and aredefined in ascending order, with respect to spatial index _(m). The zeroorder coefficient, c_(T,0,n), expresses a constant slip offset. Thefirst order coefficient, c_(T,1,n,) is a function of rotor temperature,and expresses a dynamic slip gain. The slip gain is restricted byspecific limits.

An evaluation of temperature coefficients is based on the rotortemperature, Θ_(TR,n), which can be obtained from a temperature estimateindependently provided by a thermal model, or a priori knowledge. If thetemporal index is proximate to the initial epoch of motor operation, theinitial rotor temperature 174, Θ_(TR,0), can be assumed. If rotortemperature estimation is not possible, the rated thermal operation canbe assumed by maximizing the first order temperature coefficient.

The approximate slip 138, s_(P,n), is the product of the real componentof complex input power 134 and the rated slip, with polynomial rotortemperature compensation, in (Equation 46).s _(P,n)=_(MAX)(_(MIN)(c _(T,m,0) +C _(T,m,1)·(_(REAL)(p _(F,n))·s_(0,n)),3·s ₀),0)  (Equation 46)

According to alternative aspects, the approximate slip 138 can bedetermined by other suitable methods including, but not limited to,extracting an estimate of an eccentricity frequency associated with aneccentricity harmonic.

4.3 The Mechanical Process

Referring to FIG. 8, the Mechanical Analysis process 122 defines therotor slots quantity 142, and extracts robust, accurate, transientestimates of slip 140. The Mechanical Analysis process 122 consists of aDominant Saliency Harmonic process 808, a Saliency Frequency process810, a Slip process 812, and/or a Rotor Slots Estimation process 814.

Referring to FIG. 9, a control flow 900 for the mechanical analysisprocess 122 is illustrated, where the nature of the relationshipsbetween entities and processes are defined in terms of order andconditions of operation.

The order of operation is explicitly defined by arrows, indicating thedirection of transition, from source to destination, consisting ofactors or processes. Transitions can be absolute, defined by arrowswithout text enumeration, or conditional, defined by arrows with textdefining specifically under what conditions the transition is supported.Text enumeration consists of logical statements, which can include somecombination of operators including _(AND), _(OR), and _(NOT), and datasymbols.

Collections are delimited by encapsulating rectangles, containingcollections of actors defined by rectangles, and processes defined byrounded rectangles. The epoch of control at a particular level ofabstraction is represented by a filled circle at the origin of theinitial transition. The source of the epoch transition is undefined, andnot relevant. The terminus transaction 902 is represented by a filledcircle encapsulated in an unfilled circle of larger diameter at the endof the final transaction. Precisely one epoch and one terminustransition 902 are defined per control flow.

Epoch of control flow transitions to one of three processes, dependingupon whether the rotor slots quantity 142 and/or a dominant saliencyfrequency 816 are known.

The Dominant Saliency Harmonic process 808 is selected if the rotorslots quantity 142 is known and the dominant saliency frequency 816 isunknown. The Dominant Saliency Harmonic process 808 transitions to theSaliency Frequency process 810, if the dominant saliency frequency 816is known, and otherwise transitions to the terminus of control flow 902.

The Saliency Frequency process 810 is selected if the rotor slotsquantity 142 and the dominant saliency frequency 816 are known. TheSaliency Frequency process 810 transitions to the Slip process 812. TheSlip process 812 transitions to the terminus of the control flow 902.

The Rotor Slots Estimation process 814 is selected if the rotor slotsquantity 142 is unknown. The Rotor Slots Estimation process 814transitions to the terminus of the control flow 902.

4.3.1 The Dominant Saliency Harmonic Process

Referring to FIGS. 10A and 10B, the Dominant Saliency Harmonic process808, _(DOMINANT SALIENCY HARMONIC), identifies the dominant saliencyfrequency 816, a dominant saliency order 818 and saliency filtercoefficients 404 as a function of the complex residual current 132, thefundamental frequency 136, the approximate slip 138, the rotor slots 142and the poles 172, in (Equation 47).[ƒ_(D) ,o _(D) ,{right arrow over (a)} _(H) ,{right arrow over (b)}_(H)]=_(DOMINANT SALIENCY HARMONIC)(i _(R,n),ƒ_(0,n) ,s _(P,n),R,P)  (Equation 47)ƒ_(D) Dominant Saliency Frequency 816.o_(D) Dominant Saliency Order 818.{right arrow over (a)}_(H),{right arrow over (b)}_(H) Saliency FilterCoefficients 404.i_(R,n) Complex Residual Current 132.ƒ_(0,n) Fundamental Frequency 136.s_(P,n) Approximate Slip 138.R Rotor Slots 142.P Poles 172.

Saliency harmonics present in the complex residual current 132 areidentified and evaluated to define the highest quality observable, ordominant saliency harmonic 816. The magnitude and relative proximatenoise levels of a saliency harmonic vary with motor geometry and loadconditions. Bandwidth constraints imposed by limiting the samplingfrequency to a minimally sufficient practical rate reduce the set ofobservable saliency harmonics.

A set of frequency bands are defined, centered on even integralmultiples of the fundamental frequency 136, with bandwidth equal totwice the fundamental frequency 136. It is possible for a saliencyharmonic to be observed in each frequency band in precisely oneinstance, or not at all. A unique saliency harmonic order is alsodefined for each frequency band, as a function of frequency index, orintegral fundamental frequency multiplier corresponding to the center ofthe band, and motor geometry.

The dominant saliency frequency 816 is defined as the frequency of thehighest quality observable saliency harmonic during motor operation inrated conditions. The dominant saliency order 818 corresponds to theband in which the dominant saliency harmonic exists. A dominant saliencyrange is the bandwidth of a saliency harmonic over some specificpractical range of operating conditions, which is useful in defining thecoefficients of a band limiting filter used to reduce interference insaliency frequency estimation.

The dominant saliency harmonic can be identified through the applicationof a temporal analysis method, iterating over a limited subset offrequency bands of interest. The process consists of demodulating eachcandidate saliency harmonic to extract a candidate saliency frequency.Each candidate saliency frequency is analyzed to determine whether thecorresponding candidate saliency harmonic should be excluded fromfurther consideration. The remaining candidate saliency harmonics areidentified as saliency harmonics having corresponding saliencyfrequencies 1022. The magnitude of each saliency harmonic is compared toselect the dominant saliency harmonic. Identification of the dominantsaliency harmonic results in retention of the dominant saliencyfrequency 816 and the dominant saliency order 818, and synthesis andretention of the saliency filter coefficients 404.

Complex conjugate symmetry does not apply, as the complex residualcurrent 132 is not symmetric, and no a priori knowledge is available toinfer a probability of identification of the dominant saliency harmonic,with respect to a frequency band. The resulting method is one ofexhaustive search, excluding only bands known to contain significantsupply or load related interference sources.

A demodulation process applies a VCO 602 to mix a nominal saliencyfrequency of interest 1018 in the complex residual current 132 to acomplex baseband, or a zero nominal frequency. A nominal saliencyfrequency 1018 corresponds to the expected saliency frequency duringmotor operation in rated conditions. The mixed current 1014 is bandlimited by, for example, a FIR or an IIR filter 604 to produce a complexbaseband current 424. To complete the demodulation process, a residualfrequency contained in the complex baseband current 424 is extracted,resulting in an accurate saliency frequency estimate 1022. Suitablemethods of iterative frequency estimation include, for example, Direct,PD, and PLL analysis.

Several of the bands may not contain an observable saliency harmonic,and the frequency 426 extracted from the complex baseband current 424can be considered invalid if it is not sufficiently proximate to theexpected saliency frequency associated with a specific motor 150 andload 152 condition. The Selection process 1010 identifies the dominantsaliency frequency 816 by applying, for example, a CSF filter 1008 toestimate the magnitude of each saliency harmonic.

Contiguous sequences of the complex residual current 132, thefundamental frequency 136, and the approximate slip 138 are evaluated toidentify the dominant saliency harmonic, by iteratively processing thesequences over a range of frequency bands of interest. The process isrepeated for each harmonic index, _(k), for the same sequences over adefined range. The range of the sample sequence, {right arrow over (u)},has an epoch, n, at which the approximate slip 138 exceeds a minimumthreshold, and the length of the range, N, is approximately equal to 2.0seconds, in (Equation 48).{right arrow over (u)}={[n:n+N−1]:s _(P,n)≧0.6·s ₀|_(N≧2·ƒ) _(s)}  (Equation 48)4.3.1.1 Obtaining the Complex Baseband Current

A nominal saliency order, o_(D,k), is expressed as a function of thepole quantity 172, the rotor slot quantity 142, and a harmonic index k.The harmonic index has a range in ±[12:2:30], corresponding to thefrequency bands occurring on negative and positive even integralmultiples of the normalized rated fundamental frequency, in (Equation49).

$\begin{matrix}{O_{D,k} = \left. {\left( {{k} -_{ROUND}\left( \frac{R}{0.5 \cdot P} \right)} \right) + {\,_{NOT}\left( {}_{AND}\left( {{{{k} -_{ROUND}\left( \frac{R}{0.5 \cdot P} \right)}},1} \right) \right)}} \right|_{k = {\pm {\lbrack{12:{2:30}}\rbrack}}}} & \left( {{Equation}\mspace{14mu} 49} \right)\end{matrix}$

A nominal saliency frequency 1018, f_(D,k), associated with the harmonicindex _(k), is a product of the normalized rated fundamental frequencyand a scale factor defined in terms of the rated slip, the pole quantity172, the rotor slot quantity 142, and the nominal saliency order, in(Equation 50).

$\begin{matrix}{f_{D,k} =_{SIGN}\left. {(k) \cdot \frac{f_{0}}{f_{N}} \cdot \left( {{\left( {1 - s_{0}} \right) \cdot \left( \frac{R}{0.5 \cdot P} \right)} + o_{D,k}} \right)} \right|_{k = {\pm {\lbrack{12:{2:30}}\rbrack}}}} & \left( {{Equation}\mspace{14mu} 50} \right)\end{matrix}$

A complex incident signal 1016, x_(D,k,n), is synthesized by the VCO 602at a nominal saliency frequency 1018, f_(D,k), in (Equation 51).x _(D,k,n)=_(VCO)(ƒ_(D,k))|_(k=±[12:2:30])  (Equation 51)

A complex mixed current 1014, i_(D,k,n), is formed as the product of aninstance of the conjugate of the complex incident signal 1016 and thecomplex residual current 132, in (Equation 52).i _(D,k,n) =i _(R,n) ·x* _(D,k,n)|_(k=±[12:2:30])  (Equation 52)

The saliency filter coefficients 404, {right arrow over (a)}_(H) and{right arrow over (b)}_(H), are designed with a saliency filterbandwidth, f_(B,H), equal to ½ the expected frequency range of asaliency harmonic operating over the range of normalized power in [0.0,2.0], in (Equation 53).

$\begin{matrix}{f_{B,H} = {\frac{f_{0}}{f_{N}} \cdot \frac{s_{0} \cdot R}{0.5 \cdot P}}} & \left( {{Equation}\mspace{14mu} 53} \right)\end{matrix}$

Suitable filters include, for example, linear phase filters such asvarious FIR designs with memory depth equal to approximately ¼ second,and IIR Bessel filters with comparable performance. The filterarchitecture selection is dependent largely upon processor and memoryresources, design complexity, and numerical stability. IIR and FIRfilter design techniques are diverse, well-known, and beyond the scopethis disclosure.

A designer can elect to define static coefficient sets, or dynamicallysynthesize coefficients on demand. Static coefficient sets can bedefined and analyzed a priori. IIR filters offer computationaladvantages and support compact coefficient set definitions. IIR filterstability and design for constant group delay are not trivial, butcoefficients can be designed and verified offline. FIR filters arerelatively simple to design statically or dynamically, by employingwindowing techniques, and they demonstrate trivial stability. FIR filtercoefficients are neither compact nor computationally efficient.

The Saliency filter coefficients 404 are applied to filter an instanceof the complex mixed current 1014, producing a complex baseband current424, i_(H,k,n), in (Equation 54).{right arrow over (i)} _(H,k,u)=_(IIR)({right arrow over (i)} _(D,k,u),{right arrow over (a)} _(H) ,{right arrow over (b)}_(H))|_(k=±[12:2:30])  (Equation 54)4.3.1.2 Estimating the Saliency Frequency of Each Saliency Harmonic

A demodulation process removes the nominal saliency frequency 1018 bymixing the complex residual current 132 by the conjugate of the complexexponential signal at the nominal saliency frequency 1018. The complexmixed current 1014 is band limited through application of a saliencyfilter 604 to isolate the remaining residual saliency harmonic frominterference sources, producing the complex baseband current 424. Thedemodulation process is completed by estimating the residual saliencyfrequency in the complex baseband current 424. Residual frequencyestimation extracts the instantaneous frequency of the saliencyharmonic, relative to a nominal saliency harmonic frequency 1018. Thesaliency frequency 1022 is the sum of the nominal saliency frequency1018 and the residual saliency frequency.

Alternative methods 410, m_(F), of estimating the saliency frequency1022 include, but are not limited to, Direct, PD, and PLL methods. Aspreviously described, these methods provide the flexibility to increaseprecision at the expense of computational complexity and frequencytracking rate, expanding the practical applicability of the solution. Itis contemplated that any other suitable method of estimating thesaliency frequency 1022 can be employed including those methodspreviously described with respect to estimation of the fundamentalfrequency 136.

4.3.1.2.1 The Direct Method of Estimating Each Saliency Frequency

A direct estimation of residual saliency frequency is the discretederivative of the complex baseband current phase.

A residual saliency phase, φ_(H,k,n), is the normalized phase of thecomplex baseband current 424, estimated through application of acontiguous arctangent function, in (Equation 55).

$\begin{matrix}{\varphi_{H,k,n} = {{{\,_{TAN}^{\;}\;}_{\;}^{- 1}{\left( \frac{\;_{IMAG}\left( i_{H,k,n} \right)}{\;_{REAL}\left( i_{H,k,n} \right)} \right) \cdot \frac{1}{\pi}}} + {0.5 \cdot \left( {1 -_{SIGN}\left( {}_{REAL}\left( i_{H,k,n} \right) \right)} \right) \cdot {\quad_{SIGN}\left( {}_{IMAG}\left( i_{H,k,n} \right) \right)}_{k = {\pm {\lbrack{12:{2:30}}\rbrack}}}}}} & \left( {{Equation}\mspace{14mu} 55} \right)\end{matrix}$

The normalized phase is extracted by an inverse tangent, _(TAN) ⁻¹,applied to the ratio of imaginary and real complex components, scaled bythe inverse of π to normalize the result, and adjusted to reconcile thequadrant of operation. An arctangent method can be practically definedin terms of polynomial approximation, indexed table, or some combinedmethod.

The residual saliency frequency, f_(R,k,n), is the discrete derivativeof the residual saliency phase, in (Equation 56).

$\begin{matrix}{{f_{R,k,n} = {{\frac{\mathbb{d}}{\mathbb{d}n}\left( \varphi_{H,k,n} \right)} \approx {\varphi_{H,k,n} - \varphi_{H,k,{n - 1}}}}}}_{k = {\pm {\lbrack{12:{2:30}}\rbrack}}} & \left( {{Equation}\mspace{14mu} 56} \right)\end{matrix}$

The saliency frequency 1022, f_(H,k,n), is the sum of the nominalsaliency frequency 1018 and the residual saliency frequency, in(Equation 57).ƒ_(H,k,n)=ƒ_(D,k)+ƒ_(R,k,n)≈ƒ_(D,k)+φ_(H,k,n)−φ_(H,k,n−1)|_(k=±[12:2:30])  (Equation57)

The Direct method frequency estimates are equivalent to thesuperposition of independent estimates of all complex baseband currentfrequency components, over the saliency filter bandwidth. The Directmethod forms an aggregate and biased estimate of the saliency frequency1022. In an ideal environment, a saliency harmonic can dominate thefrequency response of the complex baseband signal 424, resulting in arelatively unbiased frequency estimate. Practically, the saliency filterbandwidth generally contains significant interference which can resultin unacceptable performance in many environments.

4.3.1.2.2 The PD Method of Estimating Each Saliency Frequency

As previously discussed, the PD method offers improved accuracy,relative to the Direct method, at a cost of a modest latency and aminimal reduction in the frequency tracking rate.

The residual saliency frequency, f_(R,k,n), is estimated from theapplication of a PD to the complex baseband current 424, in (Equation58).ƒ_(R,k,n)=_(PD)(i _(H,k,n),μ_(W),α_(W))|_(k=±[12:2:30])  (Equation 58)

The saliency frequency 1022, f_(H,k,n), is the sum of nominal saliencyfrequency 1018 and residual saliency frequency, in (Equation 59).ƒf_(H,k,n)=ƒ_(D,k)+ƒ_(R,k,n)=ƒ_(D,k)+_(PD)(i_(H,k,n),μ_(W),α_(W))|_(k=±[12:2:30])  (Equation 59)4.3.1.2.3 The PLL Method of Estimating Each Saliency Frequency

Similarly, the PLL method offers improved accuracy relative to theDirect and PD methods at a further cost of latency and the frequencytracking rate. The complex baseband current 424 is a convenientrepresentation of a complex signal with zero nominal frequency. Theresidual frequency, or estimation error, of the complex baseband signal424 is estimated by a PD, and employed in adaptation of the synthesisfrequency. The residual saliency frequency estimated by the PD is usedto iteratively adapt the VCO synthesis frequency, forcing the complexmixed signal to remain at a nominal zero frequency, centered in thecomplex baseband.

A PLL filter bandwidth 2504, f_(B,p), can be defined according to thenature of the complex baseband current 424 environment. The bandwidthcan be increased, in return for significant reduction in latency andimproved frequency tracking rate, at the cost of increased aggregatefrequency estimation error. Unity bandwidth selection, which can beappropriate in environments with limited in-band interference,effectively excises the PLL IIR filter 2100, eliminating latencycontributions by the filter and maximizing frequency tracking rate.Nominal PLL filter bandwidth 2504 is 1.0.

The PLL convergence, misadjustment, frequency tracking rate, andbandwidth are dependent upon selection of the adaptive parameters, whichcan be judiciously defined to optimally support a specific environment.The nominal coefficient adaptation rate 412 and the nominal coefficientmomentum 414 are 5.0e−3 and zero, respectively. The nominal frequencyadaptation rate 418 and the nominal frequency momentum 420 are 2.0e−3and 3.5e−1, respectively.

An expected saliency frequency, f_(X,k,n), is the anticipated saliencyfrequency expressed as a function of approximate slip 138 and motorgeometry, in (Equation 60).

$\begin{matrix}{{f_{X,k,n} =_{SIGN}{(k) \cdot f_{0,n} \cdot \begin{pmatrix}{\left( {1 - s_{P,n}} \right) \cdot} \\{\left( \frac{R}{0.5 \cdot P} \right) + o_{D,k}}\end{pmatrix}}}}_{k = {\pm {\lbrack{12:{2:30}}\rbrack}}} & \left( {{Equation}\mspace{14mu} 60} \right)\end{matrix}$

The residual saliency frequency, f_(R,k,n), is estimated from theapplication of a PLL to the complex baseband current 424, initialized tothe expected residual saliency frequency and selected PLL filterbandwidth, in (Equation 61).ƒ_(R,k,n)=_(PLL)(i _(H,k,n),ƒ_(R,k,0),ƒ_(B,P) ,μ_(W),α_(W),μ_(F),α_(F))|_(k=±[12:2:30])  (Equation 61)

An initial PLL frequency, f_(R,k,0), is the expected residual saliencyfrequency, or the difference of expected saliency frequency and nominalsaliency frequency 1018, in (Equation 62).ƒ_(R,k,0)=ƒ_(X,k,0)−ƒ_(D,k)|_(k=±[12:2:30])  (Equation 62)

The saliency frequency 1022, f_(H,k,n), is the sum of nominal saliencyfrequency 1018 and residual saliency frequency, in (Equation 63).ƒ_(H,k,n)=ƒ_(D,k)+ƒ_(R,k,n)=ƒ_(D,k)+_(PLL)(i_(H,k,n),ƒ_(R,k,0),ƒ_(B,P),μ_(W),α_(W),μ_(F),α_(F))|_(k=±[12:2:30])  (Equation63)4.3.1.3 Identifying the Dominant Saliency Harmonic

If the residual saliency frequency at convergence, f_(R,k), exceeds thebounds of the saliency filter bandwidth, the potential saliencyfrequency estimate corresponding to the harmonic index, _(k), isdeclared invalid and the frequency band is eliminated from subsequentevaluation, in (Equation 64).

$\begin{matrix}\left. {f_{R,k}\underset{{\lim\mspace{14mu} m}\rightarrow u_{N - 1}}{=}\left\{ {{f_{R,k,m}} :: {{f_{R,k,m}} \leq f_{B,H}}} \right._{\begin{matrix}{k = {\pm {\lbrack{12:{2:30}}\rbrack}}} \\{N \geq {2 \cdot f_{S}}}\end{matrix}}} \right\} & \left( {{Equation}\mspace{14mu} 64} \right)\end{matrix}$

If the saliency frequency error at convergence, e_(H,k) or absolutedifference of the potential saliency frequency and the expected saliencyfrequency, exceeds a defined normalized frequency limit, the saliencyfrequency estimate corresponding to the harmonic index, _(k), isdeclared invalid and the frequency band is eliminated from subsequentevaluation, in (Equation 65).

$\begin{matrix}{e_{H,k}\underset{{\lim\mspace{14mu} m}\rightarrow u_{N - 1}}{=}\left\{ {{{{f_{H,k,m} - f_{X,k,m}}}:{{{f_{H,k,m} - f_{X,k,m}}} \leq {0.1 \cdot \frac{f_{0}}{f_{H}}}}}❘_{\begin{matrix}{k = {\pm {\lbrack{12:{2:30}}\rbrack}}} \\{N \geq {2 \cdot f_{S}}}\end{matrix}}} \right\}} & \left( {{Equation}\mspace{14mu} 65} \right)\end{matrix}$

The frequency bands are evaluated to select the dominant saliencyharmonic from the remaining candidates. Subsequent to saliency frequencyestimation and error criterion evaluation, the complex baseband current424 is filtered, for example, using a CSF filter 1008 to extract amagnitude estimate 1010 of the saliency harmonic.

A CSF filter synthesis frequency, f_(R,k), is the estimated residualsaliency frequency, evaluated at a period to exceed that needed forconvergence of the estimate, in (Equation 66).

$\begin{matrix}{f_{R,k}\underset{{\lim\mspace{14mu} m}\rightarrow u_{N - 1}}{=}{f_{R,k,m}❘_{\begin{matrix}{k = {\pm {\lbrack{12:{2:30}}\rbrack}}} \\{N \geq {2 \cdot f_{S}}}\end{matrix}}}} & \left( {{Equation}\mspace{14mu} 66} \right)\end{matrix}$

Adaptive parameters can be judiciously defined to optimally support aspecific environment. The nominal coefficient adaptation rate 418 andthe coefficient momentum 420 are 1.0e−3 and zero, respectively.

A saliency reference, {right arrow over (y)}_(H,k,u), is the CSFreference signal, operating on the complex baseband current 424, or theextracted saliency harmonic at a specific harmonic index, in (Equation67).

$\begin{matrix}{{\overset{\rightarrow}{y}}_{H,k,u} =_{CSF}{\left( {{\overset{\rightarrow}{i}}_{H,k,u},f_{R,k},\mu_{W},\alpha_{W}} \right)❘_{\begin{matrix}{k = {\pm {\lbrack{12:{2:30}}\rbrack}}} \\{N \geq {2 \cdot f_{S}}}\end{matrix}}}} & \left( {{Equation}\mspace{14mu} 67} \right)\end{matrix}$

An exponential decay filter is a 1^(st) order IIR filter withcoefficients directly specified from the exponential decay filterbandwidth, f_(B,E), in (Equation 68).

$\begin{matrix}{f_{B,E} \approx \frac{f_{0}}{f_{N}}} & \left( {{Equation}\mspace{14mu} 68} \right)\end{matrix}$

A saliency magnitude, y_(H,k), is estimated by application of anexponential decay IIR filter, in (Equation 69).

$\begin{matrix}{y_{H,k}\underset{{\lim\mspace{14mu} m}\rightarrow u_{N - 1}}{=}{{{\left( {1 - f_{B,E}} \right) \cdot {y_{H,k,{m - 1}}}} + {f_{B,E} \cdot {y_{H,k,m}}}} = {\quad_{IIR}\left( {{y_{H,k,m}},{1 - f_{B,E}},f_{B,E}} \right)}_{\begin{matrix}{k = {\pm {\lbrack{12:{2:30}}\rbrack}}} \\{N \geq {2 \cdot f_{S}}}\end{matrix}}}} & \left( {{Equation}\mspace{14mu} 69} \right)\end{matrix}$

The dominant saliency frequency 816, f_(D), is the nominal saliencyfrequency 1018 corresponding to the harmonic index, _(k), associatedwith the maximum observed saliency magnitude, in (Equation 70).

$\begin{matrix}{f_{D} = {f_{D,k}❘_{\begin{matrix}{k = {\pm {\lbrack{12:{2:30}}\rbrack}}} \\{{y_{H,k} \geq y_{H,j}},{j \neq k}}\end{matrix}}}} & \left( {{Equation}\mspace{14mu} 70} \right)\end{matrix}$

The dominant saliency order 818, O_(D), is the nominal saliency ordercorresponding to the harmonic index, _(k), associated with the dominantsaliency frequency 816, in (Equation 71).

$\begin{matrix}{o_{D} = {o_{D,k}❘_{\begin{matrix}{k = {\pm {\lbrack{12:{2:30}}\rbrack}}} \\{{y_{H,k} \geq y_{H,j}},{j \neq k}}\end{matrix}}}} & \left( {{Equation}\mspace{14mu} 71} \right)\end{matrix}$

In FIG. 11, the frequency response corresponding to each complexbaseband current is illustrated, for a 20 HP, 4 pole, 40 rotor slotsmotor operating in near rated conditions, for a period of 1.0 second, ata sampling frequency of 10 kHz. The candidate frequency responses 1102result from complex baseband currents that were produced by demodulatingthe saliency harmonic in the complex baseband current at each harmonicindex, _(k). The dominant frequency response 1104 results from theselected complex baseband current, corresponding to the dominantsaliency harmonic, which was identified at a harmonic index of 18.

FIG. 11 further depicts the complex baseband 1106 for relativecomparison, and the bandwidth of the saliency filter. The dominantsaliency frequency was identified at a normalized frequency of 0.22287,corresponding to 1114.3 Hz, or 18.572 times the rated fundamentalfrequency 166 of 60 Hz. The saliency filter bandwidth 1108 was definedover a normalized frequency range of ±2.57e−3, or ±12.8 Hz, relative tothe complex baseband. Load power is slightly higher than rated power, asthe residual saliency frequency is negative, corresponding to a highermodulation frequency than expected under nominal rated conditions, asexpressed by the dominant saliency frequency.

The selection of the highest quality observable saliency harmonicmaximizes the signal-noise ratio of the signal of interest. Therejection of potential interference sources, depicted in unfiltereddemodulated traces, is apparent.

Out-of-band interference sources can be eliminated by expressing thesaliency filter bandwidth as a function of motor geometry to correspondto the expected range of operating conditions. Advantageously, thisdynamic motor-specific interference rejection supports simplifiedsaliency frequency estimation, and improved estimation accuracy.

4.3.2 The Saliency Frequency Process

Referring to FIG. 13, the Saliency Frequency process 810,_(SALIENCY FREQUENCY), estimates the saliency frequency 426 of thedominant saliency harmonic, as a function of the complex residualcurrent 132, the dominant saliency frequency 816, and the saliencyfilter coefficients 404, in (Equation 72).ƒ_(H,n)=_(SALIENCY FREQUENCY)(i _(R,n),ƒ_(D) ,{right arrow over (a)}_(H) ,{right arrow over (b)} _(H))  (Equation 72)ƒ_(H,n) Saliency Frequency 426.i_(R,n) Complex Residual Current 132.ƒ_(D) Dominant Saliency Frequency 816.{right arrow over (a)}_(H),{right arrow over (b)}_(H) Saliency FilterCoefficients 404.

PLL initial frequency dependencies on fundamental frequency 136,approximate slip 138, dominant saliency order, rotor slots 142 and polesare not explicitly listed to improve the clarity of the description.

According to some aspects, the Saliency Frequency process 810architecture is a simplified modification of the Dominant SaliencyHarmonic process 808 architecture, consisting of the saliency frequencyestimation functionality. Synthesis frequency is constant, notiterative, and equal to the dominant saliency frequency 816, andselection is unnecessary and omitted.

A demodulation process applies a VCO 602 to mix the dominant saliencyfrequency 816 in the complex residual current 132 to a complex baseband,or a zero nominal frequency. A FIR or an IIR filter 604 is applied toband limit the mixed current 1014, producing complex baseband current424. To complete the demodulation process, the residual frequencycontained in the complex baseband current 424 is extracted, resulting inan accurate saliency frequency estimate 426. Alternative methods ofiterative frequency estimation include, for example, Direct, PD, and PLLanalysis. Saliency frequency estimation is continuous and iterative overa period of uninterrupted motor operation. Initialization andconvergence occurs subsequent to each motor start.

According to other aspects, any other suitable method for estimating thesaliency frequency can be employed including any methods previouslydescribed for estimating any frequency.

4.3.2.1 Obtaining the Baseband Current

The complex incident signal 1016, X_(D,n), is synthesized by the VCO 602at the dominant saliency frequency 816, f_(D), in (Equation 74).x _(D,n)=_(VCO)(ƒ_(D))  (Equation 74)

The complex mixed current 1014, i_(D,n), is formed as the product of theconjugate of the complex incident signal and complex residual current132, in (Equation 75).i _(D,n) =i _(R,n) ·x* _(D,n)  (Equation 75)

The saliency filter coefficients 404 are applied to filter the complexmixed current 1014, producing complex baseband current 424, i_(H,n), in(Equation 76).i _(H,n)=_(IIR)(i _(D,n) ,{right arrow over (a)} _(H) ,{right arrow over(b)} _(H))  (Equation 76)4.3.2.2 Estimating the Saliency Frequency

As described above with respect to the Dominant Saliency Harmonicprocess 808, a demodulation process removes the dominant saliencyfrequency 816 by mixing the complex residual current 132 by theconjugate of the complex exponential signal at the dominant saliencyfrequency 816. The complex mixed current 1014 is band limited throughapplication of a saliency filter 604 to isolate the remaining residualsaliency harmonic from interference sources, producing the complexbaseband current 424. Demodulation is completed by estimating theresidual saliency frequency in the complex baseband current 424. Thesaliency frequency 426 is the sum of the dominant saliency frequency 816and the residual saliency frequency.

The saliency frequency 426 can be estimated by any suitable methodincluding, but not limited to, Direct, PD, and PLL alternative methods.The Direct, PD, and PLL methods are implemented as previously describedabove with respect to the Dominant Saliency Harmonic process 808 andEquations 55-63, except saliency frequency synthesis is constant andequal to the dominant saliency frequency instead of iterative overselected potential frequencies.

4.3.2.2.1 The Direct Method of Estimating the Saliency Frequency

Thus, the residual saliency phase, φ_(H,n), is the normalized phase ofthe complex baseband current 424, estimated through application of acontiguous arctangent function, in (Equation 77).

$\begin{matrix}{\varphi_{H,k,n} = {{{\,_{TAN}^{\;}\;}_{\;}^{- 1}{\left( \frac{\;_{IMAG}\left( i_{H,k,n} \right)}{\;_{REAL}\left( i_{H,k,n} \right)} \right) \cdot \frac{1}{\pi}}} + {{0.5 \cdot \left( {1 -_{SIGN}\left( {}_{REAL}\left( i_{H,k,n} \right) \right)} \right)}{\quad{\cdot_{SIGN}\left( {}_{IMAG}\left( i_{H,k,n} \right) \right)}}_{k = {\pm {\lbrack{12:{2:30}}\rbrack}}}}}} & \left( {{Equation}\mspace{14mu} 77} \right)\end{matrix}$

The residual saliency frequency, f_(R,n), is the discrete derivative ofthe residual saliency phase, in (Equation 78).

$\begin{matrix}{f_{R,n} = {{\frac{\mathbb{d}}{\mathbb{d}n}\left( \varphi_{H,n} \right)} \approx {\varphi_{H,n} - \varphi_{H,{n - 1}}}}} & \left( {{Equation}\mspace{14mu} 78} \right)\end{matrix}$

The saliency frequency 426, f_(H,n), is the sum of dominant saliencyfrequency 816 and residual saliency frequency, in (Equation 79).

$\begin{matrix}{f_{H,n} = {{f_{D} + f_{R,n}} = {{f_{D} + {\frac{\mathbb{d}}{\mathbb{d}n}\left( \varphi_{H,n} \right)}} \approx {f_{D} + \varphi_{H,n} - \varphi_{H,{n - 1}}}}}} & \left( {{Equation}\mspace{14mu} 79} \right)\end{matrix}$4.3.2.2.2. The PD Method of Estimating the Saliency Frequency

The residual saliency frequency, f_(R,n), is estimated from theapplication of a PD to the complex baseband current 424, in (Equation80).ƒ_(R,n)=_(PD)(i _(H,n),μ_(W),α_(W))  (Equation 80)

Saliency frequency 426, f_(H,n), is the sum of dominant saliencyfrequency and residual saliency frequency, in (Equation 81).ƒ_(H,n)=ƒ_(D)+ƒ_(R,n)=ƒ_(D)+_(PD)(i _(H,n),μ_(W),α_(W))  (Equation 81)4.3.2.2.3 The PLL Method of Estimating the Saliency Frequency

The nominal PLL filter bandwidth 2504, f_(B,p), is 1.0. The nominalcoefficient adaptation rate 412 and the coefficient momentum 414 are5.0e−3 and zero, respectively. The nominal frequency adaptation rate 418and the frequency momentum 420 are 2.0e−3 and 3.5e−1, respectively.

The expected saliency frequency, f_(X,n), is the anticipated saliencyfrequency derived from approximate slip 138 and motor geometry, in(Equation 82).

$\begin{matrix}{f_{X,n} =_{SIGN}{\left( f_{D} \right) \cdot f_{0,n} \cdot \left( {{\left( {1 - s_{P,n}} \right) \cdot \left( \frac{R}{0.5 \cdot P} \right)} + o_{D}} \right)}} & \left( {{Equation}\mspace{14mu} 82} \right)\end{matrix}$

The residual saliency frequency, f_(R,n), is estimated from theapplication of a PLL to the complex baseband current 424, initialized tothe expected residual saliency frequency and the selected PLL filterbandwidth, in (Equation 83).ƒ_(R,n)=_(PLL)(i_(H,n),ƒ_(R,0),ƒ_(B,P),μ_(W),α_(W),μ_(F),α_(F))  (Equation 83)

An initial PLL frequency, f_(R,0), is the expected residual saliencyfrequency, or the difference of the expected saliency frequency and thedominant saliency frequency 816, in (Equation 84).ƒ_(R,0)=ƒ_(X,0)−ƒ_(D)  (Equation 84)

The saliency frequency 426, f_(H,n), is the sum of dominant saliencyfrequency 816 and residual saliency frequency, in (Equation 85).ƒ_(H,n)=ƒ_(D)+ƒ_(R,n)=ƒ_(D)+_(PLL)(i_(H,n),ƒ_(R,0),ƒ_(B,P),μ_(W),α_(W),μ_(F),α_(F))  (Equation 85)

Referring to FIG. 12, a transient residual saliency frequency estimationis illustrated for a synthetic complex baseband current 424 signal, fora period of 10.0 seconds, at a sampling frequency of 10 kHz.

A complex exponential signal was synthesized with a time-varyinginstantaneous frequency which increased in both amplitude and frequencywith time over a period of 5 seconds, followed by a correspondingdecrease in amplitude and frequency over a period of 5 seconds. Randomuniform noise and several complex exponential interference sources wereintroduced with increasing amplitude 5 seconds into the sequence, suchthat the noise power increased as the signal power decreased, resultingin a final signal-noise ratio of −40 dB.

The signal modulation is FM, though amplitude variation is defined whichmimics the observed increase in saliency harmonic amplitude as afunction of load 152, or input power. The synthetic residual saliencyharmonic frequency is indicative of environments with modulated saliencyharmonics, commonly observed in diverse loads 152 including, forexample, compressors or regenerative drives.

The synthetic residual saliency frequency of the source signal and theresults of the residual saliency frequency estimation performed usingthe Direct, the PD, and the PLL methods are indicated by dashed linesaccording to the legend of FIG. 12. A Fourier estimation, performed overcontiguous one second periods of observation, is also shown as a solidline. The latencies associated with the estimation methods arereconciled to support direct comparison.

The PD and PLL methods demonstrate superior accuracy relative to theDirect method, especially in the presence of interference. The Fouriermethod is unable to observe the transient behavior in the signal, as itis decidedly not stationary over the period of observation. The Fouriertransient response can only be improved at the expense of frequencyresolution, though the method is incapable of simultaneouslydemonstrating both the transient response and accuracy demonstrated bythe alternative transient estimation methods.

Referring to FIG. 14, a saliency frequency estimation is illustrated,for a 20 HP, 4 pole, 40 rotor slots motor operating in discontinuousload conditions, for a period of 5.0 seconds, at a sampling frequency of10 kHz. The Programmable Logic Control (PLC) of the load 152 wasperformed at a resolution of 20 ms. Linear step changes in load, aregenerative DC drive, were specified to occur at 2.5 second intervals.

The saliency frequency derived from an analog tachometer sensor, theresults of frequency estimation performed using Direct, PD, and PLLmethods, and the Fourier estimation, performed over contiguous onesecond periods of observation, are indicated according to the legend ofFIG. 14. The latencies associated with the estimation methods arereconciled to support direct comparison.

The Fourier saliency harmonic frequency estimation clearly depicts apattern generally corresponding to the discontinuous load conditionsspecified in PLC. It is apparent from the frequency extracted from theanalog tachometer sensor that a significant modulation of the saliencyharmonic exists, yet remains undetected by the Fourier method.

The saliency harmonic modulation has a frequency range of approximately2 Hz, and varies over that range at a modulation rate of over 3 Hz. ThePLL saliency harmonic frequency estimation method generally performswell in systems with frequency modulation rates exceeding 15 Hz, withsuperior accuracy relative to PD and Direct methods. In manyenvironments, the PD method accuracy can be comparable to the PLLmethod, while providing a higher frequency tracking rate.

The PLL, PD and Direct methods demonstrate superior transient responseand estimation accuracy, though a moderate increase in estimation noiseis evident in the Direct method.

4.3.3 The Slip Process

The Slip process 812, SLIP, estimates slip 140 as a function of thesaliency frequency 426, the dominant saliency order 818, the fundamentalfrequency 136, the rotor slots 142 and the poles 172, in (Equation 86).s _(n)=_(SLIP)(ƒ_(H,n) ,o _(D),ƒ_(0,n) ,R,P)  (Equation 86)s_(n) Slip 140.ƒ_(H,n) Saliency Frequency 426.o_(D) Dominant Saliency Order 818.ƒ_(0,n) Fundamental Frequency 136.R Rotor Slots 142.P Poles 172.

The Slip process 812 can be directly expressed though a reorganizationof the saliency frequency equation, based on availability of accuratesaliency frequency 426 and fundamental frequency 136 estimation. Thesaliency frequency 426 is estimated from the dominant saliency harmonic,and is dependent upon the corresponding dominant saliency order 818.

The saliency harmonic and the fundamental frequency 136 estimation pathsare examined to reconcile differences in latency resulting fromasymmetric processing paths. The latencies are principally contributedby the filter operations, and can readily be estimated; however, delaysassociated with adaptive elements are dependent upon adaptive parametersand call for additional analysis or experimental quantification.

The estimated slip 140, s_(n), is expressed as a function of the ratioof the saliency frequency 426 and the fundamental frequency 136, thedominant saliency order 818, the pole quantity 172, and the rotor slotquantity 142, bounded by a practical range, in (Equation 87).

$\begin{matrix}{s_{n} =_{MAX}\left( {}_{MIN}{\left( {{1 - {\left( {\frac{f_{H,n}}{f_{0,n}} - o_{D}} \right) \cdot \left( \frac{0.5 \cdot P}{R} \right)}},{3 \cdot s_{0}}} \right),0} \right)} & \left( {{Equation}\mspace{14mu} 87} \right)\end{matrix}$

Compensation for dynamic fundamental frequency 136 has proven to bedesirable in the synthesis of a robust transient estimates of slip 140.

Referring to FIG. 15, an estimate of slip 140 is illustrated, for a 20HP, 4 pole, 40 rotor slots motor operating in discontinuous loadconditions, for a period of 5.0 seconds, at a sampling frequency of 10kHz. The PLC of the load 152 was performed at a resolution of 20 ms.Linear step changes in the load 152, a regenerative DC drive, werespecified to occur at 2.5 second intervals.

The slip derived from an analog tachometer sensor, the results of slipestimation performed using Direct, PD, and PLL methods, and the Fourierestimation, performed over contiguous one second periods of observation,are indicated according to the legend of FIG. 15. The latenciesassociated with the estimation methods are reconciled to support directcomparison.

The saliency frequency modulation apparent in frequency estimation issimilarly reflected in slip 140. The PLL, the PD and the Direct methodsdemonstrate a superior transient response and estimation accuracy,though a moderate increase in estimation noise is evident in the Directmethod. The analog sensor is calibrated for rated conditions, andmeasurements are considered relative. While the frequency response ofthe sensor is sufficient to accurately represent the transient structureof the slip 140, it is likely that the PLL, the PD and the Directtransient slip estimation methods demonstrate relatively higherprecision than the sensor, over a broad range of operating conditions.

4.3.4 The Rotor Slots Estimation Process

Referring to FIG. 8, the Rotor Slots process 814, ROTOR SLOTS, estimatesthe rotor slots quantity 142, the dominant saliency frequency 816, thedominant saliency order 818, and the saliency filter coefficients 404,as a function of the complex residual current 132, the fundamentalfrequency 136, the approximate slip 138, and the poles 172, in (Equation88).[R,ƒ _(D) ,o _(D) ,{right arrow over (a)} _(H) ,{right arrow over (b)}_(H)]=_(ROTOR SLOTS)(i _(R,n),ƒ_(0,n) ,s _(P,n) ,P)  (Equation 88)R Rotor Slots 142.ƒ_(D) Dominant Saliency Frequency 816.o_(D) Dominant Saliency Order 818.{right arrow over (a)}_(H),{right arrow over (b)}_(H) Saliency FilterCoefficients 404.i_(R,n) Complex Residual Current 132.ƒ_(0,n) Fundamental Frequency 136.s_(P,n) Approximate Slip 138.P Poles 172.

The Rotor Slot process 814 extracts an estimate of static rotor slotquantity 142. The rotor slots quantity 142 can be determined by motormanufacturer data, a direct examination of the motor, or an analysis ofelectrical signals and motor parameters. Electrical analysis providesthe most practical solution for rotor slots estimation, as manufacturerdata is not readily available for all motors 150, and direct observationis intrusive, complex, and time-consuming.

The process of Rotor Slot estimation 814 is highly dependent upon theprocesses and architectures defined in the Approximate Slip process 312,the Dominant Saliency Harmonic process 808, the Saliency Frequencyprocess 810, and the Slip process 812 or alternative processes employedto determine the approximate slip 138, the dominant saliency frequency816, the estimated slip 140, or any other relevant quantities. Theapproximate slip 138 provides a reasonably accurate approximation of theactual slip, which serves as a reference signal that is independent ofmotor geometry and harmonic analysis. The dominant saliency harmonicidentification, though dependent on rotor slots 142, is iterativelyperformed over a practical range of rotor slots. The saliency frequency426 is estimated from the dominant saliency harmonic corresponding toeach rotor slots index. The estimated slip 140, directly estimated foreach saliency frequency 426, is compared with the approximate slip 138to form a performance surface, or an aggregate slip estimation error asa function of the rotor slots 142.

Rotor Slot estimation 814 is relatively complex, though it offersconsiderable opportunity for reuse, with respect to previously describedprocesses and architectures, and can be simply and clearly defined inthose terms. The rotor slots 142 describe a static quantity, based onmotor geometry, which is evaluated until an estimate is identified withconfidence. The Rotor Slots Estimation process 814 need not be boundedby hard real-time constraints, and can be practically designed toexecute offline, on a suitable previously extracted complex residualcurrent 132 sequence.

The rotor slots 142 are independently estimated from the complexresidual current 132, the fundamental frequency 136 and the approximateslip 138 sequences corresponding to some diversity of load 152 orthermal conditions before identifying the rotor slots 142 withsufficient confidence to decline further analysis. In the event that aconsensus rotor slot estimate is not identified, an optionalprobabilistic method is described to select the rotor slots 142 from aconflicting rotor slots set, based upon relative conditionalprobability.

Contiguous sequences of the complex residual current 132, thefundamental frequency 136, and the approximate slip 138 are evaluated toidentify the rotor slots 142, by iteratively processing the sequencesover a range of rotor slots of interest. The process is repeated foreach rotor slots index, _(r), for the same sequences over a definedrange. The range of the sample sequence, {right arrow over (u)}, has anepoch, n, at which the approximate slip 138 exceeds a minimum threshold,and the length of the range, N, is approximately equal to 2.0 seconds,in (Equation 89).{right arrow over (u)}={[n:n+N−1]:s _(P,n)≧·0.6·s ₀|_(N≧2·ƒ) _(s)}  (Equation 89)

Referring to FIG. 16, a control flow 1600 presents an alternative systemview, where the nature of the relationships between entities andprocesses are defined in terms of order and conditions of operation. Theorder of operation is explicitly defined by arrows, indicating thedirection of transition, from source to destination, consisting ofactors or processes. Transitions can be absolute, defined by arrowswithout text enumeration, or conditional, defined by arrows with textdefining specifically under what conditions the transition is supported.Text enumeration consists of logical statements, which can include somecombination of operators including _(AND), _(OR), and _(NOT), and datasymbols.

Collections are delimited by rectangles, and processes are defined byrounded rectangles. The epoch of control at a particular level ofabstraction is represented by a filled circle at the origin of theinitial transition. The source of the epoch transition is undefined, andnot relevant. The terminus transaction 1606 is represented by a filledcircle encapsulated in an unfilled circle of larger diameter at the endof the final transaction. Precisely one epoch and one terminustransition are defined per control flow.

The epoch of control flow 1600 transitions to the Dominant SaliencyHarmonic process 808. As previously discussed, the Dominant SaliencyHarmonic process 808 identifies the dominant saliency frequency 816 andthe dominant saliency order 818 from the complex residual current 132,the fundamental frequency 136, and the complex input power 134 signalsegments over a specified range, with the assumption that the rotorslots 142 are known, and defined by rotor slots index, _(r). If adominant saliency harmonic is not identified, the rotor slots index isincremented, and the Dominant Saliency Harmonic process 808 retainscontrol. If a dominant saliency harmonic is identified, the DominantSaliency Harmonic process 808 transitions to the Saliency Frequencyprocess 810.

The Saliency Frequency process 810 estimates the saliency frequency 426from the complex residual current 132 segment, and the dominant saliencyharmonic associated with a specific rotor slots index, and thentransitions to the Slip process 812.

The Slip process 812 estimates an estimated slip 140 from the saliencyfrequency 426, and then transitions to a Rotor Slots Performance Surfaceprocess 1602.

The Rotor Slots Performance Surface process 1602 defines an errorfunction from the difference of the approximate slip 138, derived fromthe normalized power, and the estimated slip 140, independentlyextracted from a harmonic analysis. Each execution of the Rotor SlotsPerformance Surface process 1602 produces a single error estimate, as afunction of a specific rotor slot index. Over successive iterations, aperformance surface is revealed, and a rotor slots estimatecorresponding to a global minimum is extracted. If rotor slots 142 arenot identified, and the rotor slots index is less than a practicalmaximum range, the rotor slots index is incremented, and the Rotor SlotsPerformance Surface process 1602 transitions to the Dominant SaliencyHarmonic process 808. If rotor slots 142 are identified with sufficientconfidence, the Rotor Slots Performance Surface process 1602 transitionsto the Rotor Slots process 1604.

Rotor Slots process 1604 builds a set of independent rotor slotssolutions extracted from the Rotor Slots Performance Surface process.The process is not iterative, and executes once per identified rotorslots estimate. When the set of rotor slots has sufficient density, theset is queried to determine a consensus rotor slots estimate. If noconsensus rotor slots estimate is possible, or a single rotor slotsestimate may not be extracted from the set of independent estimates withsufficient confidence, an optional probabilistic method is employed toextract a rotor slots estimate. Rotor Slots process 1604 transitions tothe terminus of control flow.

It is contemplated that according to alternative aspects, any othersuitable methods for determining the approximate slip 138, the saliencyfrequencies 426 of each rotor slot quantity, and the estimated slip 140can be utilized. For example, approximate slip 138 can be approximatedbased on an extracted estimate of an eccentricity frequency associatedwith an eccentricity harmonic.

4.3.4.1.1 Rotor Slots—the Dominant Saliency Process

The Dominant Saliency Harmonic process 808 identifies the dominantsaliency frequency 816, the dominant saliency order 818, and thesaliency filter coefficients 404 as a function of complex residualcurrent 132, the fundamental frequency 136, the approximate slip 138,the rotor slots index, _(r), and the poles 172, in (Equation 90).[ƒ_(D,r) ,o _(D,r) ,{right arrow over (a)} _(H,r) ,{right arrow over(b)} _(H,r)]=_(DOMINANT SALIENCY HARMONIC)({right arrow over (i)} _(R,u),{right arrow over (f)} _(0,u) ,{right arrow over (s)} _(P,u),r,P)|_(r=[15:100])  (Equation 90)

It is possible that a dominant saliency harmonic may not be identified.This condition occurs when no potential saliency frequency is identifiedwhich is sufficiently proximate to the expected saliency frequency tosupport verification. The cause of this condition is a particularlyinappropriate rotor slots assumption which does not agree with the motorgeometry. It is reasonable and expected that this condition willoccasionally occur, as the rotor slots index iterates over a wide rangeof potential rotor slots solutions.

Saliency filter coefficients 404 requirements can be relaxed toaccommodate practical limitations on available bandwidth and memory. Theselection of the bandwidth of the complex baseband current signal 424 tocorrespond to the expected range of the saliency frequency 426 isdependent on rotor slots 142, and optimal in terms of reducingout-of-band interference prior to residual frequency estimation.

The saliency filter coefficients 404 can be statically or dynamicallydefined to support a downsampled rotor slots range, by definingcoefficients corresponding to an integer stride of M rotor slots.

4.3.4.1.2 Rotor Slots—the Saliency Frequency Process

The Saliency Frequency process 810 estimates the saliency frequency 426of the dominant saliency harmonic, as a function of the complex residualcurrent 132, the dominant saliency frequency 816 and the saliency filtercoefficients 404, in (Equation 91).{right arrow over (f)} _(H,r,u)=_(SALIENCY FREQUENCY)({right arrow over(i)} _(R,u),ƒ_(D,r) ,{right arrow over (a)} _(H,r) ,{right arrow over(b)} _(H,r))|_(r=[15:100])  (Equation 91)4.3.4.1.3 Rotor Slots—the Slip Process

The Slip process 812 estimates slip 140 as a function of the saliencyfrequency 426, the dominant saliency order 818, the fundamentalfrequency 136, the rotor slots index, _(r), and the poles 172, in(Equation 92).{right arrow over (s)} _(r,u)=_(SLIP)({right arrow over (f)} _(H,r,u) ,o_(D,r) ,{right arrow over (f)} _(0,u) ,r,P)|_(r=[15:100])  (Equation 92)

Slip estimation is directly expressed though a reorganization of thesaliency frequency equation, based on availability of accurate saliencyfrequency and fundamental frequency estimation. Saliency frequency isestimated from the dominant saliency harmonic, and dependent upon thecorresponding dominant saliency order. The Slip process 812 is describedin section 4.3.3 above.

4.3.4.1.4 The Rotor Slots Performance Surface Process

The Rotor Slots Performance Surface process 1602 defines an errorfunction from the difference of the approximate slip 138, derived fromthe normalized power, and the estimated slip 140, independentlyextracted from a harmonic analysis. Each execution of the Rotor SlotsPerformance Surface process 1602 produces a single error estimate, as afunction of a specific rotor slot index, _(r). Over successiveiterations, a performance surface is revealed, and a rotor slotsestimate corresponding to a global minimum is extracted.

The rotor slots performance surface is a measure of a slip estimationerror as a function of rotor slots, evaluated at integer rotor slotindex, _(r), over a defined range. The rotor slots performance surfaceis nonlinear, and a concise differentiable representation for a specificmotor is unavailable, so the surface is revealed through an iterativeprocess of assuming a rotor slots definition, extracting the dominantsaliency harmonic and the slip estimations 140 derived from theassumption, and quantifying the slip estimation error.

The abscissa of the rotor slots performance surface consists of acontiguous sequence of integer rotor slot indices. The ordinate of therotor slots performance surface consists of the L1 error, or meanabsolute difference, between the approximate slip 138, and the estimateof slip 140 extracted over the rotor slots index range. The approximateslip 138 is not precise, though it can be sufficiently accurate toprovide a reference, independent of the rotor slots quantity and theslip 140.

The slip estimation error, {right arrow over (e)}_(r,u), is the absolutedifference of the slip approximation 138 and the slip 140, as a functionof rotor slots index, over the sample sequence range, {right arrow over(u)}, in (Equation 93).{right arrow over (e)} _(r,u) =|{right arrow over (s)} _(P,u) −{rightarrow over (s)} _(r,u)∥_(r=[15:100])  (Equation 93)

A sequence of the complex residual current 132 is selected anditeratively processed, over the rotor slots index range, to produce aslip estimation error signal. A single rotor slots performance surfaceis defined from the complex residual current sequence. A rotor slotsperformance surface can be queried to extract precisely one independentrotor slots estimate.

Referring to FIG. 17, the slip error estimation is illustrated, for a 15HP, 6 pole, 44 rotor slots motor operating in continuously increasing,near rated, load conditions, for a period of 2.0 seconds, at a samplingfrequency of 10 kHz. The slip approximation 138 is shown as a solid lineand the slip 140, derived from the saliency frequency 426 at thedominant saliency harmonic using a PLL, is shown as a dashed line. FIG.17 depicts the results of slip 140 from the complex residual current132, the fundamental frequency 136, and the slip approximation 138defined over a single contiguous segment for a rotor slots index equalto 44, which corresponds to the actual rotor slots quantity for thisspecific motor 150. Though the slip 140 is more precise than approximateslip 138, the temporal alignment resulting from latency compensation isdiscernable.

The region of interest, in terms of rotor slots performance surfacedefinition, is delimited by a rectangle, corresponding to the last 1.0second of the 2.0 second approximate slip 138, and estimate slip 140signals. The slip estimation error signal is defined as the absolutedifference of the approximate slip 138 and the estimate slip 140, thoughit is relevant only over the delimited region, which allows the estimateto avoid convergence and filter edge effects. Precisely one rotor slotsperformance estimate at a single rotor slots index is produced from theslip estimation error shown.

The estimate of the rotor slots performance surface corresponding to therotor slots index, 44, is equal to the mean absolute difference, or L1error, between the approximate slip 138 and the estimate slip 140, overthe delimited region of interest.

The rotor slots performance surface, ζ_(R,r), is defined iterativelyover a rotor slot index range as the mean L1 slip estimation error overa bounded post convergence period of observation with 1.0 secondduration, in (Equation 94).

$\begin{matrix}{{\zeta_{R,r} = {\frac{2}{N} \cdot {\sum\limits_{m = u_{\frac{N}{2}}}^{m = u_{N - 1}}e_{r,m}}}}}_{\begin{matrix}{r = {\lbrack{15:100}\rbrack}} \\{N \geq {2 \cdot f_{S}}}\end{matrix}} & \left( {{Equation}\mspace{14mu} 94} \right)\end{matrix}$

The rotor slots 142 are estimated by determining the global minimum ofthe rotor slots performance surface, evaluated over a practical rotorslots index range. It is possible to reduce the range of rotor slotsevaluated, if the identification of a local minimum is persistent over asufficient rotor slots range to be considered a probable global minimum.The rotor slots range reduction is computationally advantageous.

The rotor slots, R_(u), is equal to the rotor slots index, _(m), suchthat the rotor slots performance surface evaluated at this rotor slotsindex is the global minimum of the surface over a rotor slot indexrange, in (Equation 95).R _(u)={_(m):ζ_(R,m)=_(MIN)(ζ_(R))|_(r=[15:) _(MIN)_((m+P·6,100)])}  (Equation 95)

The rotor slots index range is defined from 15 rotor slots to theminimum of either 100 rotor slots, or the last rotor slots performancesurface local minimum, when traversed in increasing rotor slots indexorder, plus six times the poles 172 of the motor 150. When evaluatingthe rotor slots performance surface sequentially, evaluation can beterminated prior to the end of the rotor slots index range if a specificlocal minimum is sufficiently persistent to declare it a global minimum.

Referring to FIG. 18, the rotor slots performance surface isillustrated, for a 15 HP, 6 pole, 44 rotor slots motor operating incontinuously increasing, near rated, load conditions, for a period of2.0 seconds, at a sampling frequency of 10 kHz.

The abscissa of the rotor slots performance surface is delimited byvertical lines. The ordinate of the rotor slots performance surface isevaluated over an integer rotor slots index range in [15, 80]. The rotorslots performance surface estimates are specified by filled circles.Line segments project the linear interpolation of the rotor slotsperformance surface between the rotor slots indices, to illustrate thenonlinear nature of the surface. The rotor slots performance surfacelocal minimum 1802 is found at a rotor slots index of 44 with a slipestimation error of approximately 2.0e−4. As the local minimum 1802 ispersistent over the range of rotor slots index in [15,44+6*P], or [15,80], a global minimum is declared and the performance surface is notevaluated over the remaining rotor slots index range.

The rotor slots estimate of 44 rotor slots is correct for this specificmotor 150, based on the complex residual current 132, the fundamentalfrequency 136 and the approximate slip 138 sequences consumed inproduction of the specific rotor slots performance surface.

4.3.4.1.5 The Rotor Slots Process

The rotor slots performance surfaces can vary based on the load andthermal conditions associated with the complex residual current 132, thefundamental frequency 136 and the approximate slip 138 sequences used toproduce them. To ensure robust rotor slots estimation, several rotorslots estimates are produced from independent rotor slots performancesurfaces formed under diverse load and thermal conditions.

There is no assumption that the load 152 can be controlled, yet it cancertainly be observed. The rotor slots performance surfaces should beformed under various load conditions that are proximate to rated, as theapproximate slip 138 and the slip estimate 140 are calibrated to ratedoperation. Thermal diversity can be ensured by forming rotor slotsperformance surfaces over periodic intervals of time.

It is contemplated that any method of selecting appropriate samplesequences to produce the independent rotor slots estimates can beemployed; however, it is suggested that the normalized real input powercan be constrained to the range [0.6, 1.4]. If the input power does notvary significantly, thermal diversity can suffice to generate theindependent estimates by selecting sequences at intervals ofapproximately 1.0 hours in a continuously operating motor 150. In theevent that observable motor operation does not afford the opportunity toadhere to these goals, they can be relaxed as desired to accommodatepractical rotor slots estimation.

A set of rotor slots estimates,

, is iteratively formed as the union of independent rotor slotsestimates, based on diverse load and thermal conditions, as observable,in (Equation 96).

=

∪R _(u)  (Equation 96)

A consensus rotor slots estimate can be extracted when the rotor slotsset is sufficiently populated. If a consensus rotor slots estimate isnot available, additional independent rotor slots estimates can be addedto the set until a consensus is available, or until a defined rotorslots set population limit is reached.

The rotor slots 142, R, is defined as the consensus of the rotor slotsset, such that a supermajority of the set members are equal to the modeof the set, or in lieu of convergence, a probabilistic ProbabilityDensity Function (PDF) method, in (Equation 97).

$\begin{matrix}{R = \left\{ \begin{matrix}{{{{\,_{MODE}{()}}{\sum\limits_{R}\left( {R_{u} = {\,_{MODE}{()}}} \right)}} \geq \frac{2 \cdot M}{3}}}_{M \geq 5} \\{{{\begin{Bmatrix}{{\,_{m}{::_{PDF}\left( {P,p_{0},_{m}} \right)}} =} \\{\,_{{MA}\; X}\left( {}_{PDF}\left( {P,p_{0},} \right) \right)}\end{Bmatrix}{\sum\limits_{R}\left( {R_{u} =_{MODE}{()}} \right)}} < \frac{2 \cdot M}{3}}}_{M \geq 9}\end{matrix} \right.} & \left( {{Equation}\mspace{14mu} 97} \right)\end{matrix}$

If the rotor slots set is fully populated and a consensus rotor slotsestimate remains elusive, plurality or probabilistic methods can beemployed to define a single rotor slots estimate from the rotor slotsset.

The plurality method relaxes the consensus requirement fromsupermajority to a simple plurality of rotor slot set members to form asingle rotor slots estimate. Advantageously, the plurality method issimple and does not call for significant additional resources.

The probabilistic method extracts a specific PDF from athree-dimensional matrix of stored PDFs indexed by the poles 172 and thenormalized rated input power. The PDF, a discrete function ofprobability as a function of rotor slots index, is queried for membersof the rotor slots set. The rotor slots estimate is equal to the rotorslots set member with the highest probability. The PDF matrix issynthesized offline from a motor database, though additional memoryrequirements may not be negligible.

Referring to FIG. 20, the Rotor Slots PDF architecture is athree-dimensional matrix, indexed by poles, normalized rated inputpower, and rotor slots. As an illustration of a Rotor Slots PDF, FIG. 20represents a motor database of 5568 motors that were analyzed tosynthesize a PDF matrix 2000. The motors in the database werethree-phase induction motors with 60 Hz fundamental frequency, poles inthe set {2, 4, 6, 8, 10}, and rated input power in the range [0.5, 200]HP. The rated input power, the poles 172 and the rotor slots 142 wereknown for all motors 150. The motors 150 were separated into 5 groups,based on poles. The 10 pole group was eliminated from furtherconsideration, due to insufficient population to support meaningfulstatistical inference.

Pole groups were independently analyzed to decompose each group into 10power groups of similar rated input power. Grouping motors within eachpole group with others of similar input power led to a definition ofindependent thresholds for each group. The rated input power for motorsin a pole group was normalized to have a unity standard deviation and azero mean. An agglomerative clustering algorithm was then applied to themotors in a pole group to iteratively collect self-similar motors, interms of normalized input power, until 10 significant clusters weredefined. The normalized power bands were extracted directly from thecluster boundaries, resulting in the definition of 4 pole groups ofmotors, each consisting of 10 groups of motors segregated by normalizedrated input power.

An independent PDF was defined as the normalized histogram of rotorslots 142 for the motors 150 in each specific power group. A PDF matrixwas created by forming the PDFs into a collection indexed by poles 172and normalized rated input power bands.

Referring to FIG. 19, a PDF extraction is illustrated, for a 15 HP, 6pole motor 150, as _(PDF)(P, p₀ r), which defines the conditionalprobability of a motor 150 with poles, P, and normalized rated inputpower, p₀, as a function of rotor slots index, _(r).

The conditional probabilities for motors in this group are specified byblack circles, terminating vertical lines. Zero probabilities are notshown. A consensus rotor slots estimate 1902 of 44 rotor slots would notresult in a need to arbitrate potential conflicts in the rotor slotsset. However, if a consensus rotor slot estimation is not available, thePDF is indexed for each member of the rotor slots set, and the membercorresponding to the highest probability is assigned to the rotor slotsestimate.

The PDF illustrated in FIG. 19 indicates that though 48 rotor slots aremost probable for motors in this group, with a probability of 0.31, 44rotor slots are also common, with a probability of 0.1961. However, inthe event of a clear consensus rotor slot estimate, probability densityis irrelevant.

5.0 Component Library

The Component Library includes practical definitions for the InfiniteImpulse Response (IIR) Filter, the Voltage Controlled Oscillator (VCO),the Complex Single Frequency (CSF) Filter, the Phase Discriminator (PD),and the Phase Locked Loop (PLL).

The Component Library members support operation on complex inputs, andproduce complex outputs, when it is meaningful and appropriate to do so,unless otherwise explicitly stated. Spatial subscript m is a dimensionalindex into a vector or ordered indexed set. The spatial subscript rangeis in [0, M], where M is the highest spatial index, or order, defined. Aspatial index equal to zero corresponds to the first element, or lowestorder, in the vector or indexed set. A spatial index equal to m+1indicates the element corresponding to the next sequential position, orhigher order.

Temporal subscript n is an index into a temporal sequence, whichcontains data quantized in time at regular intervals. Temporal sequencesare generally unbounded and contiguous. A temporal index equal to zerocorresponds to the first, or original, sample in the sequence. Atemporal index equal to n+1 indicates the sample corresponding to thenext sequential quantization time, or the sample in the immediatefuture.

The Component Library descriptions include a unique functional notation,specific to the component, which supports compact and unambiguousapplication. The notation consists of one or more outputs, a functionname, in small capital letters, and a parenthesized, comma-delimitedlist of one or more inputs to the function. In the event of multipleoutputs, braces are used to delimit the output set. Spatial or temporalsequences are highlighted in bold type, and scalars are in standardtype.

5.1 The Infinite Impulse Response (IIR) Filter

Referring to FIG. 21, an IIR filter 604, _(IIR), produces a complexoutput 2106 of a linear time-invariant system in as a function of acomplex input signal 2102 and recursive coefficients 2108 and forwardcoefficients 2110, in (Equation 98).{right arrow over (y)}= _(IIR)({right arrow over (x)},{right arrow over(a)},{right arrow over (b)})  (Equation 98){right arrow over (y)} Complex Output Signal 2106.{right arrow over (x)} Complex Input Signal 2102.{right arrow over (a)} Recursive Coefficients 2108.{right arrow over (b)} Forward Coefficients 2110.

The IIR filter operation is concisely defined in terms of a discreteDifference Equation (DE), in (Equation 99). The complex output signal2106, y_(n), is expressed by superposition, as the sum of products ofprevious complex outputs and the recursive coefficients 2108, and thesum of products of previous and present inputs, x_(n), and forwardcoefficients 2110.

$\begin{matrix}{y_{n} = {{\sum\limits_{m = 1}^{M}{a_{m} \cdot y_{n - m}}} + {\sum\limits_{m = 0}^{M}{b_{m} \cdot x_{n - m}}}}} & \left( {{Equation}\mspace{14mu} 99} \right)\end{matrix}$

The transfer function, H_(z), of the IIR filter 604 is directly obtainedthrough application of the z-transform to the DE, and expressed as afunction of the complex variable z, in (Equation 100). The signconvention defined in the transfer function is strictly consistent withthe DE. The IIR filter coefficients synthesized from filter designapplications are reconciled to accommodate the transfer functiondefinition.

$\begin{matrix}{H_{Z} = {\frac{Y_{Z}}{X_{Z}} = \frac{\sum\limits_{m = 0}^{M}{b_{m} \cdot z^{- m}}}{1 - {\sum\limits_{m = 1}^{M}{a_{m} \cdot z^{- m}}}}}} & \left( {{Equation}\mspace{14mu} 100} \right)\end{matrix}$

The IIR filter order, M, is equal to the highest polynomial order of theDE. The coefficients are identified according to order by spatial index_(m). The recursive coefficient spatial indices are defined in the range[1, M], as the present output, y_(n), is not self-dependent. Forwardcoefficients indices are defined in the range [0, M]. It is possible andreasonable that the effective IIR filter order independently indicatedfrom recursive and forward coefficients may not agree. An unbalanced IIRfilter order can be reconciled by expanding the lowest ordercoefficients by appending coefficients equal to zero, such that thehighest spatial index equals the filter order, M.

A Finite Impulse Response (FIR) filter can be considered a special caseof IIR filter 604, without support for recursion. An FIR filter does notspecify recursive coefficients 2108, and analysis of the resulting DE issimplified significantly, relative to IIR filters 604. FIR filtersupport is accommodated in precisely the same manner as unbalanced IIRfilter order reconciliation, by defining recursive coefficients 2108 oflength M with coefficients equal to zero.

The recursive and forward coefficients can be complex or real. An IIRfilter 604 with a complex input signal 2102 and real coefficientsproduces a complex output signal 2106 by independently filtering thereal and complex components of the input signal. This method effectivelysubstitutes two real multiplies for each complex multiply required whencomplex coefficients are defined.

The Direct II architecture is commonly employed to implement orgraphically describe IIR filter function, as illustrated in the 2ndorder IIR filter, in FIG. 21. The Direct II architecture is derived fromreorganization of the DE, and retains direct and unmodified applicationof recursive 2108 and forward 2110 coefficients.

State Variable Analysis (SVA) is a method commonly applied to analyzelinear time-invariant systems. State variables, s_(m,n), are introducedas convenient and efficient substitutions, eliminating the need toretain and operate on input and output temporal sequences, x_(n) andy_(n). State variable spatial indices are defined in the range [1, M].SVA provides definitions for equations to solve complex output signalsynthesis, and state variable update.

Complex output signal, y_(n), is defined in terms of coefficients, a_(m)and b_(m), state variables, s_(m,n), and complex input signal, x_(n), in(Equation 101). State coefficients, c_(m), are defined in terms ofrecursive 2108 and forward 2110 coefficients as a convenientsubstitution.

$\begin{matrix}{y_{n} = {{{\sum\limits_{m = 1}^{M}{\left( {{a_{m} \cdot b_{0}} + b_{m}} \right) \cdot s_{m,n}}} + {b_{0} \cdot x_{n}}} = {{\sum\limits_{m = 1}^{M}{c_{m} \cdot s_{m,n}}} + {b_{0} \cdot x_{n}}}}} & \left( {{Equation}\mspace{14mu} 101} \right)\end{matrix}$

The state variable update specifies the next temporal iteration of thelowest order state variable, s_(1,n+1), in terms of higher order statevariables, s_(m,n), recursive coefficients, a_(m), and complex inputsignal, x_(n), in (Equation 102). Higher order next state variables,s_(m,n+1), are assigned by propagation through unit delays.

$\begin{matrix}{s_{m,{n + 1}} = \left\{ \begin{matrix}{{\sum\limits_{m = 1}^{M}{a_{m} \cdot s_{m,n}}} + x_{n}} & {m = 1} \\s_{{m - 1},n} & {m = \left\lbrack {2,M} \right\rbrack}\end{matrix} \right.} & \left( {{Equation}\mspace{14mu} 102} \right)\end{matrix}$5.2 The Voltage Controlled Oscillator (VCO)

Referring to FIG. 22, a VCO 602, _(VCO), synthesizes a complexexponential signal 2204 as a function of a synthesis frequency 2202, in(Equation 103).{right arrow over (y)}= _(VCO)({right arrow over (f)})  (Equation 103){right arrow over (y)} Complex Exponential Signal 2204.{right arrow over (f)} Synthesis Frequency 2202.

A normalized frequency is equal to the absolute frequency divided by theNyquist frequency, from (Equation 3). The synthesis frequency 2202 isintegrated, scaled, and operated on by a complex exponential poweroperation, producing a contiguous phase complex exponential signal ofunity magnitude. This process is analogous to direct FM modulation of aninput equal to the synthesis frequency.

A normalized phase, φ_(n+1), integrates normalized present phase, φ_(n),and synthesis frequency 2202, f_(n), in (Equation 104).

Phase is scaled by π and operated on by a complex exponential power,supporting direct contiguous synthesis of a complex exponential signal2204, y_(n), in (Equation 105).φ_(n+1)=φ_(n)+ƒ_(n)  (Equation 104)y _(n)=_(EXP)(j·πφ _(n))=_(COS)(π·φ_(n))+j· _(SIN)(π·φ_(n))  (Equation105)

The initial phase, φ₀, should be assigned a value equal to zero.

The phase is persistent across invocation boundaries, ensuringcontiguous operation. Phase wrapping can be carried out to ensure thatthe phase remains constrained to a practical numerical range, in [−1,1). The signed integer truncation, or floor, _(FLOOR), of the ratio ofphase and 2, is subtracted from the signed remainder, _(REM), of thephase ratio, in (Equation 106).

$\begin{matrix}{\varphi_{n + 1} =_{REM}{\left( \frac{\varphi_{n + 1}}{2} \right) -_{FLOOR}\left( \frac{\varphi_{n + 1}}{2} \right)}} & \left( {{Equation}\mspace{14mu} 106} \right)\end{matrix}$

The VCO complex exponential synthesis implementations are dependent onthe desired frequency resolution, and available dynamic range,computational complexity and memory. A simple memory based methodpredefines a table containing complex exponential function sampled at Nregular intervals over one period. A complex exponential 2204 issynthesized by defining an integer phase stride equal to the normalizedsynthesis frequency, scaled by the table length, and integrating aresidual phase error equal to the remainder of the phase stride. Theintegrated phase error is applied to the stride and reconciled to forcethe error to remain less than unity in magnitude, ensuring that nofrequency jitter is introduced due to the finite table length, orfrequency resolution. The phase jitter is inversely proportional to thetable length, or the frequency resolution.

Practical modifications to the VCO synthesis method include, but are notlimited to, storage of only cosine or sine samples, and traversing thetable with two phase indices, a cosine index, and a sine phase indexwhich follows the cosine index with a delay precisely equal to N/4, orπ/2 relative phase difference. The memory requirements can be furtherreduced at the expense of computational complexity by defining the tableover a dyadic fraction of a period, and performing region-specific indexadjustment and sign modification of the retrieved values.

5.3 The Complex Single Frequency (CSF) Filter

Referring to FIG. 23, a CSF filter 1008, CSF, produces a complexreference 2308 and a complex error signals 2306 as a function of acomplex primary signal 2304, a synthesis frequency 2302, a coefficientadaptation rate 412, and a coefficient momentum 414, in (Equation 107).[{right arrow over (y)},{right arrow over (e)}]= _(CSF)({right arrowover (d)},ƒ,μ _(W),α_(W))  (Equation 107){right arrow over (y)} Complex Reference Signal 2308{right arrow over (e)} Complex Error Signal 2306{right arrow over (d)} Complex Primary Signal 2304ƒ Synthesis Frequency 2302μ_(W) Coefficient Adaptation Rate 412 (1.0e−3)α_(W) Coefficient Momentum 414 (0)

CSF filters 1008 are high quality adaptive band-reject and band-passfilters. Their inherently superior performance, relative to staticfilter topologies, is due to the ability to dynamically estimate, ormatch, the magnitude and phase of a complex exponential component at aspecific frequency of interest in an external signal. The CSF filters1008 are computationally simple, practical, and trivially tunable to anyobservable frequency of interest.

The CSF filter 1008 consists of a VCO 602, a complex coefficient, w_(n),and a means of coefficient adaptation 2310. A complex incident signal2312, a complex reference signal 2308, and a complex error signal 2306are synthesized with respect to an external complex primary signal 2304.

Complex incident signal 2312, x_(n), is synthesized by the VCO 602 at aconstant normalized frequency 2302, f, in (Equation 108).x _(n)=_(VCO)(ƒ)  (Equation 108)

The complex reference signal 2308, y_(n), is formed as a product of theincident signal 2312 and the complex coefficient, w_(n), in (Equation109). The complex coefficient, w_(n), effects a change of magnitude andphase in the incident signal 2312, supporting synthesis of a signal thatis comparable in magnitude and coherent in phase with a complexexponential component of interest in the primary signal 2304. Thecomplex reference signal 2308 forms a narrow band pass output, withrespect to the synthesis frequency 2302.y _(n) =x _(n) ·w _(n)  (Equation 109)

The difference between the complex primary signal 2034, d_(n), and thecomplex reference signal 2308 forms the complex error signal, e_(n),which also serves as a residual, or remainder, signal, in (Equation110). The complex error signal forms a narrow band reject output, withrespect to the synthesis frequency.e _(n) =d _(n) −y _(n)  (Equation 110)

A performance surface is a measure of error as a function of coefficientspace. The CSF filter performance surface is constrained to threedimensions, real and imaginary complex coefficient dimensions andestimation error.

In a quasi-stationary environment, relative to the response of the CSFfilter 1008, an optimum complex coefficient value can be found tominimize the resulting complex error signal, in a least squares sense,resulting in synthesis of a complex reference signal 2308 thatapproximates a component of interest in the complex primary signal 2304.The optimum complex coefficient value corresponds to a global minimum ofthe performance surface. Practical complex primary signals 2304 are notstationary, resulting in an evolving performance surface, and the needto continuously evaluate the complex coefficient to minimize theestimation error.

A gradient descent is an iterative adaptive algorithm that estimates thegradient of a defined performance surface, with respect to thecoefficient dimensions, and modifies the coefficients to traverse theperformance surface in the opposite direction of the gradient estimate.If the performance surface is stationary for a sufficient period oftime, the coefficients converge about global minimum of the surface.

A Least Mean Squares (LMS) is an efficient gradient descent method whichiteratively estimates the gradient of the performance surface as afunction of estimation error.

The performance surface, ζ_(W,n), is equal to the squared error, in(Equation 111).ζ_(W,n) =e _(n) ² =e _(n) ·e* _(n)  (Equation 111)

The LMS gradient estimate, ∇_(W,n), is equal to the partial derivativeof the performance surface, with respect to the complex coefficient, in(Equation 112).

$\begin{matrix}{\nabla_{W,n}{= {\frac{\partial\zeta_{W,n}}{\partial w} = {{- 2} \cdot e_{n} \cdot x_{n}^{\star}}}}} & \left( {{Equation}\mspace{14mu} 112} \right)\end{matrix}$

The complex coefficient difference, Δ_(W,n), is equal to the differencebetween sequential complex coefficient estimates, in (Equation 113).Δ_(W,n) =w _(n) −w _(n−1)  (Equation 113)

The initial complex coefficient difference, Δ_(W,0), should be assigneda value equal to zero.

A convergence, the time to find the optimum complex coefficient, isinversely proportional to the coefficient adaptation rate 412, μ_(w). Amisadjustment, the estimation noise introduced by the adaptive process,is proportional to the coefficient adaptation rate 412. Fasterconvergence results in increased estimation noise.

The bandwidth of the CSF filter 1008 is proportional to the coefficientadaptation rate 412. This observation has implications regarding theinherent frequency resolution of the filter, and the relation of thefilter bandwidth to the range of frequency variation anticipated in thecomponent of interest in the complex primary signal 2304.

In a stationary complex primary signal environment, where the synthesisfrequency of the incident signal matches that of the instantaneousfrequency of the complex primary signal component of interest, thecomplex coefficient converges about a single stationary point tominimize error about a static complex coefficient solution.

If the synthesis frequency 2302 does not precisely match that of thecomponent of interest in the complex primary signal 2304, a staticcomplex coefficient solution is not possible. The complex coefficientcannot converge to about a single stationary point to find a minimumerror solution, as the reference and primary signals would drift apartdue to small differences in their instantaneous frequencies.

If the frequency difference is small, relative to the CSF filterbandwidth, the complex coefficient will adapt to modulate the incidentsignal to shift the instantaneous frequency of the reference signal tomatch that of the complex primary signal component of interest. Thecomplex coefficient is said to form a dynamic solution, when a staticcomplex coefficient magnitude solution rotates about the origin at afrequency equal to the difference between the complex primary componentand incident signal 2312.

Momentum is a nonlinear technique applied to improve convergence time,or the effort expended to find the optimum complex coefficient value,with potential implications on stability and misadjustment. Acoefficient momentum 414, α_(w), is applied to scale the coefficientdifference from the previous coefficient iteration, and add the productto the present iteration. The coefficient momentum 414 has a range in[0, 1).

The complex coefficient, w_(n+1), is iteratively adapted relative to thepresent complex coefficient, w_(n), by subtracting a scaled gradientestimate, and adding a momentum term, in (Equation 114).w _(n+1) =w _(n)−μ_(W)·∇_(W,n)+α_(W)·Δ_(W,n)  (Equation 114)

The initial complex coefficient, w₀, should be assigned a value equal toa small complex random number.

The dynamic nature of the complex primary signal 2304 is of principalconsideration in defining constant adaptive parameters. The coefficientadaptation rate 412 is bounded by the inverse of the largest eigenvalueof the system. The effect of momentum on stability is difficult toanalyze due to its nonlinear nature, and implicit dependence on thecoefficient adaptation rate.

5.4 The Phase Discriminator (PD)

Referring to FIG. 24, a PD 2400, _(PD), produces a frequency estimate asa function of a complex primary signal 2402, a coefficient adaptationrate 412, and a coefficient momentum 414, in (Equation 115).f= _(PD)({right arrow over (d)},μ _(W),α_(W))  (Equation 115){right arrow over (f)} Frequency 2404.{right arrow over (d)} Complex Primary Signal 2402.μ_(W) Coefficient Adaptation Rate 412. (2.0e−3).α_(W) Coefficient Momentum 414. (1.5e−1).

A PD 2400 is an adaptive filter which estimates the instantaneousfrequency 2404 of a primary signal 2402 through a process of inputnormalization, and adaptation of a complex coefficient which reconcilesthe phase difference between sequential normalized samples, encoding theinstantaneous frequency in the phase of the complex coefficient.

The PD 2400 consists of unity normalization 2406, a complex coefficient,w_(n), a means of coefficient adaptation 2408, and phase estimation2410.

A complex incident signal, x_(n), is formed as a normalized primarysignal, equal to the ratio of the complex primary signal 2402 and itsmagnitude, in (Equation 116).

$\begin{matrix}{x_{n} = \frac{d_{n}}{d_{n}}} & \left( {{Equation}\mspace{14mu} 116} \right)\end{matrix}$

Normalization 2406 of the complex primary signal 2402 preserves itsphase, while diminishing the effect of magnitude variation in sequentialsamples. Division inherent in normalization 2406 can be practicallymitigated though application of various inverse approximation algorithmsincluding, but not limited to, Newton's method.

A complex reference signal, y_(n), is formed as the product of the unitydelayed complex incident signal, x_(n−1), and a complex coefficient,w_(n), in (Equation 117).y _(n) =x _(n−1) ·w _(n)  (Equation 117)

The difference between the complex incident signal, x_(n), and complexreference signal, y_(n), forms the complex error signal, e_(n), in(Equation 118).e _(n) =x _(n) −y _(n) =x _(n) −x _(n−1) ·w _(n)  (Equation 118)

A performance surface is a measure of error as a function of coefficientspace. The PD performance surface is constrained to three dimensions,real and imaginary complex coefficient dimensions and estimation error.

The performance surface, ζ_(W,n), is equal to the squared error, in(Equation 119).ζ_(W,n) =e _(n) ² =e _(n) ·e* _(n)  (Equation 119)

The LMS gradient estimate, ∇_(W,n), is equal to the partial derivativeof the performance surface, with respect to the complex coefficient, in(Equation 120).

$\begin{matrix}{\nabla_{W,n}{= {\frac{\partial\zeta_{W,n}}{\partial w} = {{- 2} \cdot e_{n} \cdot x_{n - 1}^{\star}}}}} & \left( {{Equation}\mspace{14mu} 120} \right)\end{matrix}$

The complex coefficient difference, Δ_(W,n), is equal to the differencebetween sequential complex coefficient estimates, in (Equation 121).Δ_(W,n) =w _(n) −w _(n−1)  (Equation 121)

The initial complex coefficient difference, Δ_(W,0), should be assigneda value equal to zero.

A convergence, the time to find the optimum complex coefficient, isinversely proportional to the coefficient adaptation rate, μ_(W). Amisadjustment, the estimation noise introduced by the adaptive process,is proportional to the coefficient adaptation rate. Faster convergenceresults in increased estimation noise.

Momentum is a nonlinear technique applied to improve convergence time,or the effort expended to find the optimum complex coefficient value,with potential implications on stability and misadjustment. Coefficientmomentum, α_(W), is applied to scale the coefficient difference from theprevious coefficient iteration, and add the product to the presentiteration. Coefficient momentum has a range in [0, 1).

The complex coefficient, w_(n+1), is iteratively adapted relative to thepresent complex coefficient, w_(n), by subtracting a scaled gradientestimate, and adding a momentum term, in (Equation 122).w _(n+1) =w _(n)−μ_(W)·∇_(W,n)+α_(W)·Δ_(W,n)  (Equation 122)

The initial complex coefficient, w₀, should be assigned a value withunity magnitude and zero phase.

The complex reference signal is simply a unit delayed complex incidentsignal, scaled by a complex coefficient. As the complex incident signaland complex reference signal are normalized to unity magnitude, thecomplex error is minimized when the complex coefficient rotates thedelayed complex incident signal in phase to compensate for the phasedifference between sequential samples. The frequency is defined as phasedifference with respect to time.

The instantaneous frequency 2404 of the complex primary signal 2402 isencoded in the phase of the complex coefficient. No capability exists todiscriminate on the basis of frequency between complex exponentialcomponents in the complex primary signal 2402. An aggregateinstantaneous frequency estimate is extracted from the complex primarysignal 2402, formed from the superposition of components present in thesignal.

The frequency 2404, f_(n), of the complex primary signal 2402 is equalto the normalized phase of the complex coefficient, φ_(W,n), in(Equation 123).

$\begin{matrix}{f_{n} = {\varphi_{W,n} =_{TAN}^{- 1}{{\left( \frac{\,_{IMAG}\left( w_{n} \right)}{\,_{REAL}\left( w_{n} \right)} \right) \cdot \frac{1}{\pi}} + {0.5 \cdot \left( {1 -_{SIGN}\left( {}_{REAL}\left( w_{n} \right) \right)} \right) \cdot_{SIGN}\left( {}_{IMAG}\left( w_{n} \right) \right)}}}} & \left( {{Equation}\mspace{14mu} 123} \right)\end{matrix}$

The normalized phase is extracted by an inverse tangent, _(TAN) ⁻¹,applied to the ratio of imaginary and real complex coefficientcomponents, scaled by the inverse of π to normalize the result, andadjusted to reconcile the quadrant of operation.

The dynamic nature of the complex primary signal 2402 is of principleconsideration in defining constant adaptive parameters. The coefficientadaptation rate 412 is bounded by the inverse of the largest eigenvalueof the system. The effect of momentum on stability is difficult toanalyze due to its nonlinear nature, and implicit dependence on thecoefficient adaptation rate.

5.5 The Phase Locked Loop (PLL)

Referring to FIG. 25, a PLL 2500, _(PLL), produces a synthesis frequencyestimate 2506 as a function of a complex primary signal 2502, an initialsynthesis frequency, a filter bandwidth 2504, a coefficient adaptationrate 412, a coefficient momentum 414, a frequency adaptation rate 418,and a frequency momentum 420, in (Equation 124).{right arrow over (ƒ)}=_(PLL)({right arrow over (d)},ƒ₁,ƒ_(B,P),μ_(W),α_(W),μ_(F),α_(F))  (Equation 124){right arrow over (f)} Synthesis Frequency 2506.{right arrow over (d)} Complex Primary Signal 2502.ƒ₁ Initial Synthesis Frequency.ƒ_(B,p) Filter Bandwidth 2504. (1.0).μ_(W) Coefficient Adaptation Rate 412. (5.0e−3).α_(W) Coefficient Momentum 414. (0).μ_(F) Frequency Adaptation Rate 418. (2.0e−3).α_(F) Frequency Momentum 420. (3.5e−1).

The PLL 2500 is a closed loop adaptive filter optimally suited toaccurately estimate instantaneous frequency in a dynamic environmentwith significant in-band interference. Adaptive frequency synthesis andinterference rejection support the identification and tracking of acomplex exponential component of interest in a complex primary signal.

The PLL 2500 consists of a VCO 602, a mixer, an IIR filter 2100, a PD2400, and a means of frequency adaptation 2508. A VCO 602 synthesizes acomplex exponential signal at an instantaneous frequency of interest.The product of the conjugate of the complex exponential signal and thecomplex primary signal is band limited with an IIR filter 2100,resulting in a complex baseband signal. The complex baseband is aconvenient representation of a complex signal with zero nominalfrequency. The residual frequency, or estimation error, of the complexbaseband signal is estimated by a PD 2400, and employed in adaptation ofthe synthesis frequency 2508.

The complex incident signal, x_(n), is synthesized by the VCO at adynamic normalized frequency, f_(n), in (Equation 125).x _(n)=_(VCO)(ƒ_(n))  (Equation 125)

The complex mixed signal, y_(n), is formed as a product of the conjugateof the carrier signal and the primary signal, in (Equation 126).y _(n) =x* _(n) ·d _(n)  (Equation 126)

Frequency tracking rate is the maximum rate at which the instantaneousfrequency of the complex exponential component of interest in thecomplex primary signal 2502 can change, while the PLL 2500 contiguouslytracks the component without a significant increase in estimation error,or residual frequency. If the rate of change of frequency in thecomponent exceeds the tracking rate of the PLL, estimation error willnot be reduced to a level sufficient to represent convergence.

The complex baseband signal, y_(B,n), is equal to the complex mixedsignal, band limited by application of an IIR filter 2100, in (Equation127). The IIR filter 2100 reduces interference and improves thefrequency estimation accuracy, at the expense of the frequency trackingrate and latency.y _(B,n)=(1−ƒ_(B,P))·y _(B,n−1)+ƒ_(B,P) ·y _(n)=_(IIR)(y_(n),1−ƒ_(B,P),ƒ_(B,P))  (Equation 127)

An exponential decay filter is a 1st order IIR filter 2100 withcoefficients directly specified from the filter bandwidth 2504, f_(B,p).The effective memory depth is inversely proportional to filter bandwidth2504. The filter bandwidth 2504 can be increased, in return forsignificant reduction in the latency and improved frequency trackingrate, at the cost of increased aggregate frequency estimation error.

The filter bandwidth selection is proportional to the expected frequencyrange of the complex exponential signal of interest in the complexprimary signal 2502. As the filter bandwidth 2504 and the tracking rateare inversely proportional, the dynamic nature of the component ofinterest is considered in assignment of filter bandwidth 2504. Unitybandwidth selection, which can be appropriate in environments withlimited interference, effectively excises the IIR filter 2100,eliminating latency contributions by the filter and maximizing frequencytracking rate.

The residual frequency, f_(R,n), of the complex baseband signal isestimated by the PD 2400, in (Equation 128).ƒ_(R,n)=_(PD)(y _(B,n),μ_(W),α_(W))  (Equation 128)

Residual frequency is the error basis for adaptation of the synthesisfrequency 2508. PLL convergence is attained, in a stationaryenvironment, when the synthesis frequency is equal to the instantaneousfrequency of the complex exponential component of interest in theprimary signal 2502, and the residual frequency is approximately zero,neglecting estimation noise introduced by the adaptive process.

A performance surface is a measure of error as a function of coefficientspace. The PLL performance surface is constrained to two dimensions,synthesis frequency 2506 and estimation error.

Direct expression and differentiation of the performance surface as afunction of synthesis frequency 2506 is indirectly dependent on theadaptive complex coefficient in the PD 2400. It is often convenient todirectly express the gradient estimate as a simplified approximation,without formal definition of a differentiable performance surface.

The LMS gradient estimate, ∇_(F,n), is equal to the partial derivativeof the performance surface, with respect to the estimation error, in(Equation 129). Residual frequency provides a convenient and sufficientapproximation to the gradient.

$\begin{matrix}{\nabla_{F,n}{= {\frac{\partial\zeta_{F,n}}{\partial f} \approx {- f_{R,n}}}}} & \left( {{Equation}\mspace{14mu} 129} \right)\end{matrix}$

The residual frequency difference, Δ_(F,n), is equal to the differencebetween sequential residual frequency estimates, in (Equation 130). Thecomplex coefficient difference assignment is forced to zero when thesign of the difference changes, relative to the previous iteration.

$\begin{matrix}{\Delta_{F,n} = \left\{ \begin{matrix}0 & {{\,_{SIGN}\left( {f_{R,n} - f_{R,{n - 1}}} \right)} \neq {\,_{SIGN}\left( \Delta_{F,{n - 1}} \right)}} \\{f_{R,n} - f_{R,{n - 1}}} & {{\,_{SIGN}\left( {f_{R,n} - f_{R,{n - 1}}} \right)} = {\,_{SIGN}\left( \Delta_{F,{n - 1}} \right)}}\end{matrix} \right.} & \left( {{Equation}\mspace{14mu} 130} \right)\end{matrix}$

The initial residual frequency difference, Δ_(F,0), should be assigned avalue equal to zero.

Convergence, the time to find the optimum synthesis frequency, isinversely proportional to the frequency adaptation rate, μ_(F).Misadjustment, the estimation noise introduced by the adaptive process,is proportional to the frequency adaptation rate. Faster convergenceresults in increased estimation noise.

Momentum is a nonlinear technique applied to improve convergence time,or the effort expended to find the optimum synthesis frequency value,with potential implications on stability and misadjustment. Frequencymomentum, α_(F), is applied to scale the coefficient difference from theprevious coefficient iteration, and add the product to the presentiteration. Frequency momentum has a range in [0, 1).

The synthesis frequency, f_(n+1), is iteratively adapted relative to thepresent synthesis frequency, f_(n), by subtracting a scaled gradientestimate, and adding a momentum term, in (Equation 131).ƒ_(n+1)=ƒ_(n)−μ_(F)·∇_(F,n)+α_(F)·Δ_(F,n)  (Equation 131)

The initial synthesis frequency, f₀, should be assigned an applicationspecific value. A priori knowledge of the approximate frequency of acomplex exponential signal component of interest in the complex primarysignal 2502 can be applied to improve convergence, and to ensure thatthe appropriate component is successfully locked, or isolated andtracked, by the PLL.

The dynamic nature of the complex primary signal is of principleconsideration in defining constant adaptive parameters. The coefficientadaptation rate 412 and the frequency adaptation rate 418 are bounded bythe inverse of the largest eigenvalues of their respective systems. Theeffect of momentum on stability is difficult to analyze due to itsnonlinear nature, and implicit dependence on adaptation rate.

To ensure convergence, frequency should change slowly, relative tomagnitude and phase estimations implicit in complex coefficientadaptation. Therefore, the frequency adaptation rate 418 can be lessthan the coefficient adaptation rate 412.

While the present invention has been described with reference to one ormore particular embodiments, those skilled in the art will recognizethat many changes can be made thereto without departing from the spiritand scope of the present invention. Each of these embodiments andobvious variations thereof is contemplated as falling within the spiritand scope of the claimed invention, which is set forth in the followingclaims.

1. A method of determining a slip estimate associated with an inductionmotor through analysis of voltage and current signals, the methodcomprising: determining a fundamental frequency from a representation ofthe voltage signal; determining a saliency frequency from arepresentation of the current signal; determining an estimation of aslip quantity according to a slip estimation function that includes thesaliency frequency, a saliency order, the fundamental frequency, aquantity of rotor slots, and a quantity of poles of the motor; andstoring the estimation of the slip quantity.
 2. The method of claim 1,further comprising selecting a saliency harmonic associated with thesaliency frequency.
 3. The method of claim 1, wherein the determiningthe fundamental frequency includes filtering the representation of thevoltage signal to reduce interference outside of a range of frequenciesthat includes a rated fundamental frequency associated with the motor.4. A method of determining a slip estimate associated with a motor, themethod comprising: determining a fundamental frequency from arepresentation of a voltage signal; determining a saliency frequencyfrom a representation of a current signal; determining an estimation ofa slip quantity according to a slip estimation function that includesthe saliency frequency, a saliency order, the fundamental frequency, aquantity of rotor slots, and a quantity of poles of the motor; andstoring the estimation of the slip quantity, wherein the representationof the voltage signal is a complex voltage signal, and wherein thedetermining the fundamental frequency includes: determining a complexincident signal including a complex exponential evaluated at a synthesisfrequency related to a rated fundamental frequency associated with themotor; determining a complex mixed voltage according to a function thatincludes multiplying the complex voltage signal by a representation ofthe complex incident signal; determining a complex baseband voltageaccording to a function that includes filtering the complex mixedvoltage; determining a residual fundamental frequency according to afunction that at least approximates a derivative of a phase of thecomplex baseband voltage; and determining a function that includes thesynthesis frequency and the residual fundamental frequency to producethe fundamental frequency.
 5. The method of claim 4, wherein thefunction for determining the residual fundamental frequency includes anoperation that at least approximates a representation of a derivative ofa phase of the complex baseband voltage.
 6. The method of claim 4,wherein the function for determining the residual fundamental frequencyis carried out via a complex Phase Locked Loop (PLL) comprising:adaptively determining the residual fundamental frequency according to afunction that includes the complex baseband voltage, wherein the complexbaseband voltage is a complex primary signal of the PLL, and an initialsynthesis frequency of the PLL by: determining a complex incident signalof the PLL including a complex exponential evaluated at a synthesisfrequency of the PLL, according to a complex Voltage ControlledOscillator (VCO) function; determining a complex mixed signal of the PLLaccording to a function that includes multiplying the complex primarysignal of the PLL by the conjugate of the complex incident signal of thePLL; determining a complex baseband signal of the PLL according to afunction that includes filtering the complex mixed signal of the PLL;determining a residual frequency of the PLL according to a function thatat least approximates a representation of a derivative of a phase of thecomplex baseband signal of the PLL; modifying the synthesis frequency ofthe PLL according to a function that includes the synthesis frequency ofthe PLL and the residual frequency of the PLL; and determining afunction that includes a representation of the synthesis frequency ofthe PLL to produce the residual fundamental frequency.
 7. The method ofclaim 6, further comprising adding a representation of the ratedfundamental frequency associated with the motor and the residualfundamental frequency to produce the fundamental frequency.
 8. Themethod of claim 4, wherein the function for determining the residualfundamental frequency includes a complex Phase Discriminator (PD). 9.The method of claim 8, wherein the complex PD includes: determining acomplex incident signal of the PD by normalizing each sample of thecomplex baseband signal such that the resulting sequence of complexsamples of the complex incident signal has the same magnitude, whereinthe complex incident signal of the PD retains the phase of the complexbaseband signal; equating a complex reference signal of the PD to theproduct of a delayed representation of the complex incident signal ofthe PD and a complex coefficient; determining a complex error signal ofthe PD according to a function that includes the difference between thecomplex incident signal of the PD and the complex reference signal ofthe PD; modifying the complex coefficient according to a function thatincludes the complex coefficient and the complex error signal of the PD,such that the complex error signal of the PD is adaptively minimized asthe modified complex coefficient at least approximates a derivative ofthe complex incident signal of the PD; and defining the residualfundamental frequency according to a function that includes a phase ofthe complex coefficient.
 10. The method of claim 1, further comprisingdetermining an approximate slip according to an approximate slipfunction.
 11. The method of claim 10, wherein the approximate slipfunction includes: determining a normalized real input power accordingto an input power function that includes the representation of thevoltage signal, the representation of the current signal, a ratedfundamental frequency associated with the motor, and a rated input powerassociated with the motor; determining a rated slip associated with themotor according to a function that includes a rated speed associatedwith the motor and the quantity of poles; and multiplying the normalizedreal input power and the rated slip associated with the motor.
 12. Themethod of claim 11, wherein the input power function includes:extracting, from a complex voltage of the representation of the voltagesignal, a complex fundamental voltage such that a fundamental frequencycomponent of the complex voltage is retained; extracting, from a complexcurrent of the representation of the current signal, a complexfundamental current such that a fundamental frequency component of thecomplex current is retained; and multiplying the complex fundamentalvoltage with the conjugate of the complex fundamental current.
 13. Themethod of claim 11, further comprising temperature compensating theapproximate slip according to a function that includes multiplying theapproximate slip by a coefficient that is related to the temperature ofthe motor.
 14. The method of claim 10, wherein the approximate slipfunction includes extracting an estimate of an eccentricity frequencyassociated with an eccentricity harmonic, wherein the determining theapproximate slip is carried out according to a function that includesthe eccentricity frequency, an eccentricity order, and the quantity ofpoles.
 15. The method of claim 1, wherein the representation of thevoltage signal is complex, and the representation of the current signalis complex.
 16. The method of claim 1, wherein the determining thesaliency frequency includes filtering the representation of the currentsignal to reduce interference outside of a range of frequencies thatincludes a nominal saliency frequency associated with the motor.
 17. Themethod of claim 16, wherein the nominal saliency frequency is calculatedaccording to a function that includes a rated fundamental frequencyassociated with the motor, a rated slip associated with the motor, anominal saliency order, the quantity of rotor slots, and the quantity ofpoles.
 18. A method of determining a slip estimate associated with amotor, the method comprising: determining a fundamental frequency from arepresentation of a voltage signal; determining a saliency frequencyfrom a representation of a current signal; determining an estimation ofa slip quantity according to a slip estimation function that includesthe saliency frequency, a saliency order, the fundamental frequency, aquantity of rotor slots, and a quantity of poles of the motor; andstoring the estimation of the slip quantity, wherein the representationof the current signal is a complex current signal, and wherein thedetermining the saliency frequency includes: determining a complexincident signal including a complex exponential evaluated at a synthesisfrequency related to a nominal saliency frequency associated with themotor; determining a complex mixed current according to a function thatincludes multiplying the complex current signal by a representation ofthe complex incident signal; determining a complex baseband currentaccording to a function that includes filtering the complex mixedcurrent; determining a residual saliency frequency according to afunction that at least approximates a derivative of a phase of thecomplex baseband current; and determining a function that includes thesynthesis frequency and the residual saliency frequency to produce thesaliency frequency.
 19. The method of claim 18, wherein the function fordetermining the residual saliency frequency includes an operation thatat least approximates a representation of a derivative of a phase of thecomplex baseband current.
 20. The method of claim 18, wherein thefiltering the complex mixed current is carried out via a filter having afilter bandwidth that is defined according to a function that includes arated fundamental frequency associated with the motor, a rated slipassociated with the motor, the quantity of rotor slots, and the quantityof poles.
 21. The method of claim 18, wherein the function fordetermining the residual saliency frequency is carried out via a complexPhase Locked Loop (PLL) comprising: adaptively determining the residualsaliency frequency according to a function that includes the complexbaseband current, wherein the complex baseband current is a complexprimary signal of the PLL, and an initial synthesis frequency of the PLLby: determining a complex incident signal of the PLL including a complexexponential evaluated at a synthesis frequency of the PLL, according toa complex Voltage Controlled Oscillator (VCO) function, determining acomplex mixed signal of the PLL according to a function that includesmultiplying the complex primary signal of the PLL by the conjugate ofthe complex incident signal of the PLL, determining a complex basebandsignal of the PLL according to a function that includes filtering thecomplex mixed signal of the PLL, determining a residual frequency of thePLL according to a function that at least approximates a representationof a derivative of a phase of the complex baseband signal of the PLL,and modifying the synthesis frequency of the PLL according to a functionthat includes the synthesis frequency of the PLL and the residualfrequency of the PLL; and determining a function that includes arepresentation of the synthesis frequency of the PLL to produce theresidual saliency frequency.
 22. The method of claim 21, furthercomprising adding a representation of the nominal saliency frequency andthe residual saliency frequency to produce the saliency frequency. 23.The method of claim 21, wherein the saliency order is a nominal saliencyorder and wherein the initial synthesis frequency of the PLL is definedaccording to a function that includes the nominal saliency frequency, arated fundamental frequency associated with the motor, a slip associatedwith the motor, the nominal saliency order, the quantity of rotor slots,and the quantity of poles.
 24. The method of claim 18, wherein thefunction for determining the residual saliency frequency includes acomplex Phase Discriminator (PD).
 25. The method of claim 24, whereinthe complex PD includes: determining a complex incident signal of the PDby normalizing each sample of the complex baseband signal such that theresulting sequence of complex samples of the complex incident signal hasthe same magnitude, wherein the complex incident signal of the PDretains the phase of the complex baseband signal; equating a complexreference signal of the PD to the product of a delayed representation ofthe complex incident signal of the PD and a complex coefficient;determining a complex error signal of the PD according to a functionthat includes the difference between the complex incident signal of thePD and the complex reference signal of the PD; modifying the complexcoefficient according to a function that includes the complexcoefficient and the complex error signal of the PD, such that thecomplex error signal of the PD is adaptively minimized as the modifiedcomplex coefficient at least approximates a derivative of the complexincident signal of the PD; and defining the residual saliency frequencyaccording to a function that includes a phase of the complexcoefficient.